Skew mirrors, methods of use, and methods of manufacture

ABSTRACT

An optical reflective device referred to as a skew mirror, having a reflective axis that need not be constrained to surface normal, is described. Examples of skew mirrors are configured to reflect light about a constant reflective axis across a relatively wide range of wavelengths. In some examples, a skew mirror has a constant reflective axis across a relatively wide range of angles of incidence. Exemplary methods for making and using skew mirrors are also disclosed. Skew mirrors include a grating structure, which in some examples comprises a hologram.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application claims priority from co-pending U.S. Application Nos.62/209,290, filed 24 Aug. 2015, and titled “MULTIWAVELENGTH DIFFRACTIONGRATING MIRRORS, METHODS OF USE, AND METHODS OF MANUFACTURE,” and62/318,917, filed 6 Apr. 2016, and titled “SKEW MIRRORS, METHODS OF USE,AND METHODS OF MANUFACTURE.” The above applications are incorporatedherein by reference, in their entireties.

FIELD OF THE INVENTION

The present invention relates generally to optical reflective devicescomprising grating structures.

BACKGROUND

Conventional dielectric mirrors are produced by coating a surface(typically glass) with layers of materials that differ from each otherin their electric permittivity. The layers of materials are typicallyarranged so that Fresnel reflections from layer boundaries reinforceconstructively, producing large net reflectivity. Broadband dielectricmirrors can be designed by ensuring that this condition obtains over arelatively broad specified range of wavelengths and incidence angles.However, because the layers are deposited on a surface, the reflectiveaxis of a dielectric mirror is necessarily coincident with surfacenormal, i.e. the reflective axis is perpendicular to the mirror surface.Because of this constraint on the reflective axis, a dielectric mirroris disposed in some devices in a configuration that is suboptimal forpurposes other than reflection. Similarly, the reflective axis beingconstrained to surface normal makes a dielectric mirror entirelyinadequate for some purposes. Moreover, glass dielectric mirrors tend tobe relatively heavy, making them suboptimal or inappropriate forapplications requiring a relatively lightweight reflective component.

Conversely, conventional grating structures can reflect light about areflective axis that differs from surface normal of the medium in whichthe grating structure resides. However, for a given angle of incidence,angles of reflection for conventional grating structures typicallyco-vary with wavelength of incident light. Thus, using a conventionalgrating structure to reflect light avoids the constraint inherent indielectric mirrors that reflective axis coincide with surface normal.However, where a substantially constant reflective axis is required, aconventional grating structure is substantially limited to a singlewavelength (or very narrow range of wavelengths) for a given angle ofincidence. Similarly, a conventional grating structure is limited to asingle angle of incidence (or very narrow range of incidence angles), inorder to reflect light of a specified wavelength about a constantreflective axis.

Accordingly, requirements for a relatively simple device that reflectslight about a reflective axis not constrained to surface normal, andwhose angle of reflection for a given angle of incidence is constant atmultiple wavelengths, are not met by currently available reflectivedevices comprising either reflective grating structures or dielectricmirrors. A need therefore exists for such a reflective device, and suchneed may be acute in head mounted display devices.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a cross-section view illustrating reflective properties of askew mirror according to an embodiment.

FIG. 1B is a cross-section view illustrating reflective properties of askew mirror according to an embodiment.

FIG. 2A is a cross-section view illustrating reflective properties of askew mirror according to an embodiment.

FIG. 2B is a cross-section view illustrating reflective properties of askew mirror according to an embodiment.

FIG. 3 is a cross-section view of a system for making a skew mirror,according to an embodiment.

FIG. 4 is a cross-section view illustrating a method of making a skewmirror, according to an embodiment.

FIG. 5A is a cross-section view of a hologram recorded in a gratingmedium.

FIG. 5B is a cross-section view of a k-space representation of a singlesinusoidal hologram.

FIG. 6A is a cross-section view of a k-space representation of a singlesinusoidal hologram.

FIG. 6B cross-section view of a k-space representation of a singlesinusoidal hologram.

FIG. 7 is a cross-section real view illustrating reflective propertiesof a skew mirror in real space, according to an embodiment.

FIG. 8A is a cross-section view of a k-space representation of a skewmirror according to an embodiment.

FIG. 8B is a cross-section view of a k-space representation of a skewmirror according to an embodiment.

FIG. 9A is a cross-section view of a k-space representation of a skewmirror according to an embodiment.

FIG. 9B is a cross-section view of a k-space representation of a skewmirror according to an embodiment.

FIG. 10A is a cross-section view illustrating reflective properties of askew mirror according to an embodiment.

FIG. 10B is a cross-section view of a k-space representation of a skewmirror according to an embodiment.

FIG. 10C is a cross-section view of a k-space representation of a skewmirror according to an embodiment.

FIG. 10D is a cross-section view of a k-space representation of a skewmirror according to an embodiment.

FIG. 11A is a cross-section view of a k-space representation of a skewmirror according to an embodiment.

FIG. 11B is a cross-section view of a k-space representation of a skewmirror according to an embodiment.

FIG. 12A is a cross-section view illustrating reflective properties of askew mirror according to an embodiment.

FIG. 12B is a cross-section view illustrating reflective properties of askew mirror according to an embodiment.

FIG. 12C is a cross-section view illustrating reflective properties of askew mirror according to an embodiment.

FIG. 13A is a cross-section view illustrating reflective properties of askew mirror according to a wave guide embodiment.

FIG. 13B is a cross-section view illustrating reflective properties of askew mirror according to an embodiment.

FIG. 14A is a cross-section view of a k-space representation of a skewmirror according to an embodiment.

FIG. 14B is a cross-section view illustrating reflective properties of askew mirror according to an embodiment.

FIG. 15 is a plan view illustrating reflective properties of a skewmirror according to an embodiment.

FIG. 16A is a cross-section view illustrating a system for making a skewmirror, according to an embodiment.

FIG. 16B is a cross-section view illustrating a system for making a skewmirror, according to an embodiment.

DETAILED DESCRIPTION

Embodiments of the present invention include a reflective devicecomprising a grating medium within which resides a hologram or othergrating structure. The grating medium, by virtue of the gratingstructure residing therein, has physical properties that allow it todiffract light about an axis, referred to as a reflective axis, whereinangle of diffraction (henceforth referred to as angle of reflection) issubstantially constant, (i.e. it varies by less than) 1° for multiplewavelengths of light incident upon the grating medium at a given angleof incidence. In some embodiments, the above phenomenon is observed formultiple angles of incidence.

Similarly, embodiments typically have a substantially constantreflective axis across a range of incidence angles for incident light ofa given wavelength, and this phenomenon may be observed with incidentlight at various wavelengths. In some embodiments, the reflective axisremains substantially constant for every combination of a set ofmultiple incidence angles and a set of multiple wavelengths

In some embodiments, the grating structure includes a hologram generatedby interference between multiple light beams referred to as recordingbeams. Typically, but not necessarily, the grating structure includesmultiple holograms. The multiple holograms may be recorded usingrecording beams incident upon the grating medium at angles that varyamong the multiple holograms, and/or using recording beams whosewavelengths vary among the multiple holograms. In some embodiments, thegrating structure includes a hologram recorded using two recording beamswhose angles of incidence upon the grating medium vary while thehologram is being recorded, and/or whose wavelengths vary while thehologram is being recorded. Embodiments further include a device whereinthe reflective axis differs from surface normal of the grating medium byat least 1.0 degree; or at least by 2.0 degrees; or at least by 4.0degrees; or at least by 9.0 degrees.

A First Embodiment Skew Mirror

A first embodiment skew mirror 100 is illustrated in FIGS. 1A and 1B.The first embodiment skew mirror 100 comprises a grating structure 105(shown by diagonal hatch lines in FIGS. 1A and 1B) residing in a gratingmedium 110. For purposes of clarity, the diagonal hatch lines areomitted in a region within the grating medium 110 proximate figureelements indicating light, axes, and angles. However, persons skilled inthe art will recognize that the grating structure 105 typically occupiesthe region described above. The grating structure 105 of the firstembodiment includes multiple holograms that at least partially spatiallyoverlap with each other in the grating medium 110.

The multiple holograms are recorded into the grating medium internalvolume and thus extend below the grating medium surface 112.Accordingly, they are sometimes referred to as volume holograms. Themultiple holograms of the first embodiment comprise forty eight (48)volume holograms, recorded with recording beams having a wavelength of405 nm. Each of the 48 volume holograms typically at least partiallyspatially overlaps all others of the 48 volume holograms in the gratingmedium 110. In some embodiments, each of the multiple holograms at leastpartially spatially overlaps at least one, but not all, of the other ofthe multiple holograms. Recording the 48 holograms of the firstembodiment skew mirror is described below in a first method of making askew mirror. In some embodiments, the grating structure includes between1 and 48 holograms; or between 4 and 25 holograms; or at least 5holograms; or at least 9 holograms; or at least 11 holograms; or atleast 24 holograms.

The first embodiment grating medium 110 is a proprietary photosensitivepolymeric optical recording medium, designated AK174-200, available fromAkonia Holographics, LLC (Longmont, Colo.). The AK174-200 recordingmedium of the first embodiment is approximately 200 um thick, has an M/#of approximately 18, and a refractive index of approximately 1.50 for405 nm light. Optical recording mediums such as the AK174-200 medium area type of grating medium in which grating structures can be recorded byoptical means. Grating mediums are typically, but not necessarily, atleast 70 um thick to approximately 1.2 mm thick. The AK174-200 mediumtypically undergoes relatively little shrinkage (usually about 0.1% to0.2%) as a result of recording volume holograms. Variations of gratingmediums include, but are not limited to, photorefractive crystals,dichromated gelatin, photo-thermo-refractive glass, and film containingdispersed silver halide particles.

Variations of the first embodiment skew mirror 100 may include anadditional layer such as a glass cover or glass substrate (not shown inFIGS. 1A and 1B). The additional layer may serve to protect the gratingmedium from contamination, moisture, oxygen, reactive chemical species,damage, and the like. The additional layer is typically refractive indexmatched to the grating medium 110. Because the refractive index for theadditional layer is usually very close to the refractive index of thegrating medium, refraction of light at the interface of the additionallayer and the grating medium can usually be ignored. For the firstembodiment, refractive indices for both the additional layer and thegrating medium are approximately 1.5 for light having a wavelength of405 nm. For clarity, the additional layer is not shown in FIGS. 1A and1B.

As best seen in FIG. 1A, the grating structure 105 of the firstembodiment has the physical property of being configured to reflect afirst incident light 124A, 124B, about a first reflective axis 138(shown in broken line). The first incident light consists essentially ofa collimated, monochromatic light beam. The first incident lightfurthermore includes a first wavelength of 532 nm and is incident uponthe grating medium 110 at a specific site 117. The first reflective axis138 differs from surface normal 122 of the grating medium by a firstreflective axis angle 135 of +13.759 degrees (internal, relative tosurface normal), where the first incident light has an first internalangle of incidence 125A, 125B relative to surface normal, from −4.660degrees (shown as first incident light 124A) to +1.933 degrees (shown asfirst incident light 124B), resulting in a range of 6.593 degrees. Thefirst internal angles of incidence for the first incident light includeone hundred (100) different internal angles spaced at angle intervals ofabout 0.067 degrees, from −4.660 degrees to +1.933 degrees, as shown intable A-1 in Appendix A. In some variations of the first embodiment skewmirror, the first internal angles of incidence for the first incidentlight include ten (10) different internal angles spaced at angleintervals of about 0.67 degrees, from −4.660 degrees to +1.933 degrees.Throughout this specification and appended claims, identified angles andangle values refer to internal angles relative to surface normal, unlessclearly indicated otherwise.

As shown FIG. 1A, first incident light 124A, having a first internalangle of incidence of 125A of −4.660 degrees relative to surface normal,is reflected by the grating structure 105 as first reflected light 127A,having a first internal angle of reflection 126A of +32.267 degreesrelative to surface normal. First incident light 124B, having a firstinternal angle of incidence 125B relative to surface normal of +1.933degrees, is reflected as first reflected light 127B having a firstinternal angle of reflection 126B of +25.668 degrees. First reflectedlight 127A, 127B has the first wavelength, i.e. in the first embodimentthe first reflected light has a wavelength of 532 nm. First incidentlight angles, first reflected light angles, and first reflective axisangles for the first embodiment skew mirror are shown in Table A-1,appended to this specification in Appendix A.

Incident light and its reflection are bisected by the reflective axissuch that the internal angle of incidence of the incident light relativeto the reflective axis has the same magnitude as the internal angle ofreflection of the reflected light relative to the reflective axis. Thusit can be said that the incident light and its reflection exhibitbilateral symmetry about the reflective axis.

As best seen in FIG. 1B, the grating structure 105 of the firstembodiment is further configured to reflect second incident light 130A,130B about a second reflective axis 139. The second incident lightconsists essentially of a collimated, monochromatic light beam. Thesecond incident light furthermore includes a second wavelength of 513 nmand is incident upon the grating medium 110 at the specific site 117.The specific site 117 includes an area of the grating medium surface 112upon which both the first and second incident light shine. The secondreflective axis 139 differs from surface normal 122 of the gratingmedium by a second reflective axis angle 136 of +13.693 degrees(internal) relative to surface normal, where the second incident lighthas a second internal angle of incidence, relative to surface normal,from −4.660 degrees to +1.933 degrees. The second internal angle ofincidence includes one hundred (100) different internal angles spaced atangle intervals of approximately 0.067 degrees, from −4.660 degrees to+1.933 degrees. In some variations of the first embodiment skew mirror,the second internal angles of incidence for the second incident lightinclude ten (10) different internal angles spaced at angle intervals ofabout 0.67 degrees, from −4.660 degrees to +1.933 degrees.

As shown in FIG. 1B, second incident light 130A, having a secondinternal angle of incidence 128A of −4.660 degrees relative to surfacenormal, is reflected by the grating structure 105 as second reflectedlight 133A, having a second internal angle of reflection 133A of +32.075degrees relative to surface normal. Second incident light 130B, having asecond internal angle of incidence 128B relative to surface normal of+1.933 degrees, is reflected as second reflected light 133B having asecond internal angle of reflection 129B of +25.273 degrees. Secondreflected light 133A, 133B has the second wavelength, i.e. in the firstembodiment the second reflected light has a wavelength of 513 nm. Secondincident light angles, second reflected light angles, and secondreflective axis angles for the first embodiment skew mirror, are shownin Table A-2, appended to this specification in Appendix A.

The first wavelength (532 nm) differs from the second wavelength (513nm) by 19 nm, which can be represented by a value referred to as a wavefraction (WF), defined as

${{WF} = \frac{\left( {{\lambda\; 1} - {\lambda\; 2}} \right)}{\left( {{\lambda\; 1} + {\lambda\; 2}} \right)/2}},$where λ1=a longer wavelength among multiple wavelengths, and λ2=ashorter wavelength among the multiple wavelengths. Thus where themultiple wavelengths consist of a first wavelength of 532 nm and asecond wavelength of 513 nm,

${WF} = {\frac{\left( {532 - 513} \right)}{\left( {532 + 513} \right)/2} = 0.036}$Similarly, where the multiple wavelengths consist of a continuousspectrum from 390 nm or less to at least 700 nm, WF≥0.57. Embodimentsinclude, but are not limited to, variations in which WF≥0.005; WF≥0.010;WF≥0.030; WF≥0.10; WF≥0.250; WF≥1.0; or WF≥2.0. The wave fraction (WF)defined by a longer (λ1) and shorter (λ2) wavelengths in the rangetypically, but not necessarily, includes a continuous spectrum ofwavelengths between λ1 and λ2.

The second reflective axis angle 136 differs from the first reflectiveaxis angle 135 by 0.0661 degree. Accordingly, the second reflective axisis substantially coincident with the first reflective axis, meaning thatthe second reflective axis angle 136 differs from first reflective axisangle 135 by 1.0 degree or less. Such small difference betweenreflecting axis angles across a range of wavelengths (in this case,across a WF of 0.039) can be a necessity where a nondispersive mirror isrequired. For some applications, the difference between reflective axisangles should be 0.250 degree or less for WF=0.030. Similarly, for someother applications, the difference between reflective axis angles shouldbe equal 0.10 degree or less for WF=0.030.

Relative to the first reflective axis, internal angles of incidence ofthe first incident light vary from −11.867 degrees to −18.464 degrees.Relative to the second reflective axis, internal angles of incidence ofthe second incident light vary from −11.670 degrees to −18.368 degrees.Thus it can be said that each of the first incident light and secondincident light is offset from the first reflective axis by at least11.670 degrees. In embodiments, incident light may be offset from itsreflective axis by an internal angle of at least 1.0 degree; by at least2.0 degrees; by at least 5.0 degrees; or by at least 9.0 degrees. A skewmirror or other reflective device configured to reflect incident lightthat is offset from the incident light's reflective axis can beadvantageous in some applications. For example, in a head mounteddisplay it may be advantageous to reflect an image toward a user's eye,but not to retroreflect the image back toward its source. Suchreflection toward a user's eye typically requires that incident light beoffset from its reflective axis by an internal angle of at least 5.0degrees, and more typically by at least 9.0 degrees. Similarly, a deviceutilizing total internal reflection typically requires that incidentlight be offset from its reflective axis.

First embodiment external angles relative to surface normal for incidentlight and its reflection are also illustrated in FIGS. 1A and 1B. Asseen in FIG. 1A, external angles relative to surface normal for firstincident light 124A, 124B ranges from first incident light externalangle 113A of −7.000 degrees to first incident light external angle 113Bof +2.900 degrees. As seen in FIG. 1B, external angles relative tosurface normal for second incident light 130A, 130B ranges from secondincident light external angle 115A of −7.000 to second incident lightexternal angle 115B of +2.900 degrees. First reflected light externalangles 114A, 114B and second reflected light external angles 116A, 116Bare also illustrated in FIGS. 1A and 1B, respectively. External anglesare measured with the skew mirror residing in air, with refractionoccurring at the skew mirror/air boundary. Angles of incidence andangles of reflection, and reflective axis angles are tabulated in TablesA-1 and A-2 of Appendix A.

The physical properties of the first embodiment allow it to reflectlight having other wavelengths, and to reflect light incident upon thegrating medium at other angles. For example, the first embodimentgrating structure's reflective properties allow it to reflect lighthaving a wavelength of 520.4 nm about a reflective axis having a meanreflective axis angle of +13.726 degrees that varies by 0.10 degree orless where angles of incidence of the 520.4 nm light range from −6.862degrees to +13.726 degrees and all angles in between, for a range of20.588 degrees. In another example of its reflective properties, thefirst embodiment is configured to reflect incident light about areflective axis (having a mean reflective axis angle of)+13.726° thatvaries by 0.20 degree or less for all wavelengths from 503 nm to 537 nm(a range of 34 nm, WF=0.065, including a continuous spectrum ofwavelengths between 503 nm and 537 nm), where the angle of incidence(internal, relative to surface normal) is −1.174 degrees.

For clarity, light in FIGS. 1A and 1B is illustrated as being reflectedat a point residing proximate a center of the grating structure 105.However, persons skilled in the art recognize that light is typicallyreflected throughout the grating structure rather than at a specificpoint.

In some embodiments, the first incident light and the second incidentlight have wavelengths other than 532 and 513, respectively. Similarly,embodiments include first and second reflective axes that may becoincident with surface normal, or may differ from surface normal.

A Second Embodiment Skew Mirror

A second embodiment skew mirror 200 is illustrated in FIGS. 2A and 2B.The second embodiment skew mirror 200 comprises a grating structure 205(shown by diagonal hatch lines in FIGS. 2A and 2B) residing in a gratingmedium 210. For purposes of clarity, the diagonal hatch lines areomitted in a region within the grating medium 210 proximate figureelements indicating light, axes, and angles. However, persons skilled inthe art will recognize that the grating structure 205 typically occupiesthe region described above. The grating structure 205 of the secondembodiment includes multiple holograms that at least partially overlapwith each other in the grating medium 210. The multiple holograms of thesecond embodiment comprise forty nine (49) volume holograms, recordedwith recording beams having a wavelength of 405 nm. The 49 volumeholograms overlap each other in the grating medium 210, and are recordedin a manner similar to the first embodiment skew mirror, except thatrecording beam internal angles of incidence are adjusted to account formedia shrinkage. Recording the 49 holograms of the second embodimentskew mirror is described below in a second method of making a skewmirror.

The second embodiment grating medium 210 is a proprietary photosensitivepolymeric optical recording medium, designated AK233-200, available fromAkonia Holographics, LLC (Longmont, Colo.). The AK233-200 recordingmedium of the second embodiment is approximately 200 um thick, has anM/# of approximately 24, and a refractive index of approximately 1.50for light having a wavelength of 405 nm. The AK233-200 medium typicallyshrinks about 0.50% as a result of recording volume holograms.

Variations of the second embodiment skew mirror 200 may include anadditional layer such as a glass cover or glass substrate (not shown inFIGS. 2A and 2B). The additional layer is typically refractive indexmatched to the grating medium, and a thin film of index matching fluidmay reside between the grating medium 210 and the additional layer.

As best seen in FIG. 2A, the grating structure 205 of the secondembodiment has the physical property of being configured to reflect afirst incident light 224A, 224B, about a first reflective axis 238(shown in broken line). The first incident light of the secondembodiment consists essentially of a collimated, monochromatic lightbeam. The first incident light furthermore includes a first wavelengthof 532 nm and is incident upon the grating medium 210 at a specific site217. The first reflective axis 238 differs from surface normal 222 ofthe grating medium by a first reflective axis angle 235 of +14.618degrees (internal) relative to surface normal, where the first incidentlight has a first internal angle of incidence 225A, 225B, relative tosurface normal, residing between −9.281 degrees to −2.665 degrees,inclusive (a range of 6.616 degrees). The first internal angle ofincidence includes one hundred one (101) different internal anglesspaced at angle intervals of approximately 0.066 degrees, from −9.281degrees to −2.665 degrees. In some variations of the second embodimentskew mirror, the first internal angles of incidence for the firstincident light include ten (10) different internal angles spaced atangle intervals of about 0.66 degrees, from −9.281 degrees to −2.665degrees.

As shown FIG. 2A, first incident light 224A, having a first internalangle of incidence 225A of −9.281 degrees relative to surface normal, isreflected by the grating structure 205 as first reflected light 227A,having a first internal angle of reflectance 226A of +38.610 degreesrelative to surface normal. First incident light 224B, having a firstinternal angle of incidence 225B relative to surface normal of −2.665degrees, is reflected as first reflected light 227B having a firstinternal angle of reflectance 226B of +31.836 degrees. First reflectedlight 224A, 224B has the first wavelength, i.e. in the second embodimentthe first reflected light has a wavelength of 532 nm. First incidentlight angles, first reflected light angles, and first reflective axisangles, for the second embodiment skew mirror are shown in Table A-3,appended to this specification in Appendix A.

As best seen in FIG. 2B, the grating structure 205 of the secondembodiment is further configured to reflect second incident light 230A,230B about a second reflective axis 239. The second incident light ofthe second embodiment consists essentially of a collimated,monochromatic, light beam. The second incident light furthermoreincludes a second wavelength of 513 nm, and the second wavelengththerefore differs from the first wavelength by 19 nm, or a wave fraction(WF) of 0.036. The second incident light is incident upon the gratingmedium 210 at the specific site 217. The specific site 217 of the secondembodiment includes an area of the grating medium surface 212 upon whichboth the first and second incident light shine. The second reflectiveaxis 239 differs from surface normal 222 of the grating medium by asecond reflective axis angle 236 of +14.617 degrees (internal) relativeto surface normal, where the second incident light has a second internalangle of incidence 228A, 228B relative to surface normal, spanning arange of −9.281 degrees to −2.665 degrees. The second internal angle ofincidence of the second incident light includes one hundred one (101)different internal angles spaced at angle intervals of approximately0.066 degrees, from −9.281 degrees to −2.665 degrees. In some variationsof the second embodiment skew mirror, the second internal angles ofincidence for the second incident light include ten (10) differentinternal angles spaced at angle intervals of about 0.66 degrees, from−9.281 degrees to −2.665 degrees.

As shown in FIG. 2B, second incident light 230A, having a secondinternal angle incidence 228A of −9.281 degrees relative to surfacenormal, is reflected by the grating structure 205 as second reflectedlight 233A, having a second internal angle of reflectance 229A of+38.598 degrees relative to surface normal. Second incident light 230B,having a second internal angle of incidence 228B relative to surfacenormal of −2.655 degrees, is reflected as second reflected light 233Bhaving a second internal angle of reflectance 229B of +31.836 degrees.Second reflected light 233A, 233B has the second wavelength, i.e. in thesecond embodiment the second reflected light has a wavelength of 513 nm.Second incident light angles, second reflected light angles, and secondreflective axis angles for the second embodiment skew mirror 200 areshown in Table A-4, appended to this specification in Appendix A.

For clarity, light in FIGS. 2A and 2B is illustrated as being reflectedat a point residing proximate a center of the grating structure 205.However, persons skilled in the art recognize that light is typicallyreflected throughout the grating structure rather than at a specificpoint.

In the second embodiment, the second reflective axis angle differs fromthe first reflective axis angle by approximately 0.0005 degree acrossWF=0.036. This very low level of change can approach the level ofprecision of instrumentation used to measure reflection angles.Accordingly, for the purposes of the present invention, the secondreflective axis can be said to not differ from the first reflectiveaxis. For some applications, the difference between reflective axisangles should be 0.025 degree or less. For some other applications, thedifference between reflective axis angles should be 0.010 degree or lessacross WF≥0.036. The second embodiment skew mirror meets theserequirements. A Student's t-test (two-tailed) indicates no differencebetween the first reflective axis angle and the second reflective axisangle (N=101 per group; P=0.873). Moreover, a difference of 0.001 degreeor less challenges the precision of instrumentation used to measure skewmirror reflection angles. Accordingly, for purposes of the presentinvention, where a second reflective axis differs from a firstreflective axis by 0.001 degree or less, the second reflective axis canbe said to not differ from the first reflective axis.

For the second embodiment skew mirror, angles of incidence of the firstincident light vary from −17.250 degrees to −23.946 degrees relative tothe first reflective axis. Angles of incidence of the second incidentlight relative to the second reflective axis vary from −17.250 degreesto −23.940 degrees. Thus it can be said that each of the first incidentlight and second incident light is offset from the first reflective axisby at least 17.20 degrees. For the second embodiment skew mirror, anglesif incidence and angles of reflection relative to reflective axis, forincident light and its reflection, respectively, are tabulated in TablesA-3 and A-4 of Appendix A.

Second embodiment external angles relative to surface normal forincident light and its reflection are also illustrated in FIGS. 2A and2B. As seen in FIG. 2A, external angles relative to surface normal forfirst incident light 224A, 224B ranges from first incident lightexternal angle 213A of −14.000 degrees to first incident light externalangle 213B of −4.000 degrees. As seen in FIG. 2A, external anglesrelative to surface normal for second incident light 230A, 230B rangesfrom second incident light external angle 215A of −14.000 to secondincident light external angle 215B of −4.000 degrees. First reflectedlight external angles 214A, 214B and second reflected light externalangles 216A, 216B are also illustrated in FIGS. 2A and 2B, respectively.

Persons skilled in the art will recognize that incident light and itsreflection can typically be reversed, such that what was previously anangle of reflection becomes and angle of incidence, and vice versa.However, for purposes of the present invention, recitation ordescription of incidence angles refers only to those incidence anglesbeing oriented to one side of the incidence angles' reflective axes, or,in the case of retroreflected incident light, an incidence angle of zero(0) relative to the reflective axis. Accordingly, a range of incidenceangles does not include angles that are both positive and negative withrespect to the reflective axes. As illustrated and described here,incidence angles are negative (i.e. in a clockwise direction) withrespect to the incident lights' reflective axes. However, thisconvention is used for convenience and simplicity and is not meant toteach, suggest, or imply that a skew mirror can only reflect lightresiding to one side of a reflective axis.

A Third Embodiment Skew Mirror

A third embodiment skew mirror comprises a grating structure residing ina grating medium, wherein the grating structure comprises twenty one(21) volume holograms that overlap each other in the grating medium.

The third embodiment grating medium is a commercial photosensitivepolymeric optical recording medium, designated BAYFOL® HX TPphotopolymer film, available from Covestro AG (formerly BayerMaterialScience AG) (Leverkusen, Germany). The BAYFOL® HX TP recordingmedium of the third embodiment is approximately 70 um thick, andtypically shrinks about 1.0% as a result of recording volume holograms.Accordingly, shrinkage compensation is typically employed when recordingvolume holograms in the third embodiment grating medium. Shrinkagecompensation is described below in the method of making the thirdembodiment skew mirror.

Variations of the third embodiment skew mirror may include an additionallayer such as a glass cover or glass substrate. The additional layer istypically refractive index matched to the grating medium, and a thinfilm of index matching fluid may reside between the third embodimentgrating medium and the additional layer.

The grating structure of the third embodiment has the physical propertyof being configured to reflect a first incident light about a firstreflective axis. The first incident light has a first wavelength of 532nm and is incident upon the grating medium at a specific site. The firstreflective axis differs from surface normal of the grating medium by afirst reflective axis angle of +9.419 degrees (internal) relative tosurface normal, where the first incident light has an internal angle,relative to surface normal, residing between −6.251 degrees and +0.334degrees, inclusive (a range of 6.585 degrees). The internal angle of thefirst incident light includes multiple angles spanning a range ofapproximately 6.59 degrees, the multiple angles including one hundred(100) different internal angles spaced at angle intervals ofapproximately 0.067 degrees, from −6.251 degrees to +0.334 degrees.

Third embodiment first incident light having an internal angle of −6.251degrees relative to surface normal, is reflected by the gratingstructure as first reflected light having an internal angle of +25.027degrees relative to surface normal. First incident light having aninternal angle relative to surface normal of +0.334 degrees is reflectedas first reflected light having an internal angle of +18.487 degrees.First reflected light has the first wavelength, i.e. in the thirdembodiment the first reflected light has a wavelength of 532 nm.

The grating structure of the third embodiment is further configured toreflect second incident light about a second reflective axis. The secondincident light has a second wavelength of 513 nm, and the secondwavelength therefor differs from the first wavelength by 19 nm, or awave fraction (WF) of 0.036. The second incident light is incident uponthe grating medium at the specific site. The second reflective axisdiffers from surface normal of the grating medium by a second reflectiveaxis angle of +9.400 degrees (internal) relative to surface normal,where the second incident light has in internal angle, relative tosurface normal, spanning a range from −6.251 degrees to +0.334 degrees.The internal angle of the second incident light includes one hundred(100) different internal angles spaced at angle intervals ofapproximately 0.067 degrees, from −6.251 degrees to +0.334 degrees.

Third embodiment second incident light, having an internal angle of−6.251 degrees relative to surface normal, is reflected by the gratingstructure as second reflected light, having an internal angle of +24.967degrees relative to surface normal. Second incident light having aninternal angle relative to surface normal of +0.334 degrees is reflectedas second reflected light having an internal angle of +18.425 degrees.Second reflected light has the second wavelength, i.e. in the thirdembodiment the second reflected light has a wavelength of 513 nm. Thesecond reflective axis of the third embodiment is substantiallycoincident with the first reflective axis.

Tables 1 includes a summary of reflective properties of first, second,and third embodiment skew mirrors.

TABLE 1 DIFFERENCE BETWEEN REFLECTIVE AXIS ANGLES AT λ = 532 nm AND λ =513 nm FIRST SECOND THIRD EMBODIMENT EMBODIMENT EMBODIMENT SKEW MIRRORSKEW MIRROR SKEW MIRROR (AK174-200 (AK233-200 (BAYFOL ® HX recordingmedium) recording medium) recording medium) N = 100 N = 101 N = 100measurements measurements measurements Mean reflective axis 13.693°14.617° 9.400° INTERNAL angle at λ = 532 nm* Mean reflective axis13.759° 14.618° 9.419° INTERNAL angle at λ = 513 nm* Difference between0.066° 0.0005° 0.018° reflective axis INTERNAL angle at λ = 532 nm andat λ = 513 nm** Incident Light −4.660° to +1.933° −9.281° to −2.665°−6.251° to +0.334° INTERNAL Angles*** (range = 6.593°) (range = 6.616°)(range = 6.585°) Mean reflective axis 22.234° 25.594° 14.720° EXTERNALangle at λ = 532 nm* Mean reflective axis 22.110° 25.593° 14.690°EXTERNAL angle at λ = 513 nm* Difference between 0.124° 0.0005° 0.030°reflective axis EXTERNAL angle at λ = 532 nm and at λ = 513 nm**Incident Light −7.000° to 2.900° −14.000° to −4.000° −9.400° to +0.501°EXTERNAL Angles*** *mean angles are relative to surface normal, and arethe means of N measurements at N incident light angles of incidence;both incident and reflected light have the specified wavelength (λ).**differences between mean reflective axis angles at λ = 532 nm and at λ= 513 nm are absolute values and thus excludes negative numbers.***incident light angles of incidence, relative to surface normal.Methods of Making a Skew Mirror

An exemplary system 350 for making a skew mirror is illustrated in FIG.3. The exemplary system 350 includes a grating medium 310 disposedbetween a first mirror 352A and a second mirror 352B. The first andsecond mirrors are arranged to direct a first recording beam 354 and asecond recording beam 355 such that the recording beams intersect andinterfere with each other to form an interference pattern that isrecorded as a hologram 305 in the grating medium 310. The hologram 305is an example of a grating structure.

The recording beams may be referred to as a reference beam and a signalbeam according to a convention sometimes used by persons skilled in theholographic arts. However, each of the first and second recording beamsare typically monochromatic collimated plane wave beams that areidentical to each other (except for angles at which they are incidentupon the grating medium). Moreover, the so-called signal beam typicallyincludes no data encoded therein that is not also present in theso-called reference beam. Thus designation of one recording beam as asignal beam and the other recording beam as a reference beam can bearbitrary, with the designation of “signal” and “reference” serving todistinguish between the two recording beams, rather than to indicatethat the one recording beam includes encoded data not present in theother recording beam. In some embodiments the recording beams may havewidths that differ from each other.

The grating medium 310 is typically secured in place between a firstprism 359A and second prism 359B using a fluid index matched to both theprisms and the grating medium. A skew axis 361 resides at a skew angle364 relative to surface normal 322. The first and second recording beams354, 355 reside at a first recording beam internal angle 356 and asecond recording beam internal angle 357, respectively, relative surfacenormal 322. As can be seen in FIG. 3, the first and second recordingbeams 354, 355 are symmetrical about the skew axis 361 such that thefirst recording beam internal angle relative to the skew axis 366 isequal to 180° minus the second recording beam internal angle relative tothe skew axis 367. The internal angles of the first and second recordingbeams relative to the skew axis 366, 367 are readily calculated from thefirst and second recording beam internal angles 356, 357, respectively,and the skew angle 364.

Each of the first and second recording beams are typically collimatedplane wave beams originating from a laser light source. The plane wavebeams may be illustrated using multiple light ray depictions for eachrecording beam. For clarity however, in FIG. 3 the first and secondrecording beams are illustrated using a single light ray depiction foreach recording beam.

Refraction at air/prism boundaries, for example where the firstrecording beam 354 intersects an air/prism boundary of the first prism359A and where the second recording beam 355 intersects an air/prismboundary of the second prism 359B, is shown figuratively rather thanstrictly quantitatively in FIG. 3. Because the prisms are typicallyindex matched to the grating medium 310, refraction at the prism/gratingmedium boundary can usually be ignored. In embodiments, the gratingmedium and prisms each have an index of refraction of approximately1.50.

For purposes of the present invention, a skew angle can be substantiallyidentical to a reflective axis angle, meaning the skew angle is within1.0 degree of the reflective axis angle. Persons skilled in the art willrecognize that the skew axis angle and reflective axis angle can betheoretically identical. However, due to limits in system precision andaccuracy, shrinkage of recording medium that occurs during recordingholograms, and other sources of measurement error, the skew angle asmeasured or estimated based on recording beam angles may not perfectlymatch the reflective axis angle as measured by incidence angles andreflection angles of light reflected by a skew mirror. Nevertheless, askew angle determined based on recording beam angles can be within 1.0degree of the reflective axis angle determined based on angles ofincident light and its reflection, even where medium shrinkage andsystem imperfections contribute to errors in estimating skew angle andreflective axis angle. A skew axis/reflective axis is generally called askew axis when referring to making a skew mirror (for example whendescribing recording a hologram in a skew mirror grating structure), andas a reflective axis when referring to light reflective properties of askew mirror.

Angles at which the first and second recording beams 354, 355 areincident upon the grating medium are adjusted by rotating the first andsecond beam mirrors, 352A, 352B, respectively. Rotation of the beammirrors, indicated by rotation arrows 353, not only adjusts incidenceangles, but also would change where the recording beams intersect thegrating medium 310. Accordingly, the grating medium 310 and prisms 359A,359B are moved translationally in order to record holograms atapproximately the same location in the grating medium. Translation ofthe grating medium 310 is indicated by translation arrow 360.

In a variation of the exemplary system 350, a variable wavelength laseris used to vary the wavelength of the first and second recording beams.Incidence angles of the first and second recording beams may be, but arenot necessarily, held constant while the wavelength of the first andsecond recording beams is changed.

A First Method of Making a Skew Mirror

A first method of making a skew mirror is illustrated in FIG. 4. Theskew mirror of the first method is the first embodiment skew mirror 100,which is also illustrated in FIGS. 1A and 1B, and whose physicalproperties are described above. The first method typically utilizes asystem for making a skew mirror such as the exemplary system 350illustrated in FIG. 3 and described above. For clarity however, in FIG.4 first and second prisms are omitted, and recording beams areillustrated without showing refraction at air/grating medium boundariesor air/prism boundaries. However, persons skilled in the art willrecognize that refraction typically occurs at an air/prism boundary (orair/grating medium boundary, where index matched prisms are not used),and should be accounted for when designing a system or method to achievethe internal angles described.

A first recording beam 154 and a second recording beam 155 are directedat the first embodiment grating medium 110, where the recording beamsinterfere with each other to create an interference pattern, which isrecorded as a volume hologram in the grating medium 110. The recordingbeams are typically created by splitting a 405 nm light beam from anexternal cavity, tunable diode laser into two separate beams. The lightbeam is split using a polarizing beam splitter, and a half wave plate isused to alter polarity of one of the two separate beams from p-polarizedto s-polarized, such that both of the two separate beams ares-polarized. One of the s-polarized beams becomes the first recordingbeam 154 and the other of the s-polarized beams becomes the secondrecording beam 155. Each of the first and second recording beams is acollimated, plane wave beam having a wavelength of 405 nm.

The first embodiment skew mirror benefits from having reflectiveproperties that allow it to reflect light at a substantially differentwavelength, and in particular a considerably longer wavelength, than therecording beam wavelength. The AK174-200 grating medium, in which firstembodiment holograms are recorded with 405 nm wavelength recordingbeams, absorbs 405 nm light at approximately 0.07 absorbance units forthe 200 um thick medium. Conversely, the AK174-200 grating medium hasnegligible absorbance for visible wavelengths of light greater than 425nm (conservatively estimated at less than 0.002 absorbance units per 200um; the negligible absorbance is typically indistinguishable from zero).Thus the AK174-200 grating medium absorbs recording beam light (at 405nm) at least 35 times more strongly than green light (for example, in arange of 503 nm to 537 nm) the first embodiment skew mirror isconfigured to reflect.

The grating structure 105 of the first embodiment skew mirror 100 iscreated by recording 48 volume holograms in the grating medium 110. Eachof the 48 holograms is recorded at its own unique first recording beaminternal angle 156 and its own unique second recording beam internalangle 157. The first recording beam internal angle 156 is an internalangle of the first recording beam 154 relative to surface normal 122 ofthe grating medium 110 and the second recording beam internal angle 157is an internal angle of the second recording beam 155 relative tosurface normal 122. Each of the first and second recording beams for thefirst embodiment skew mirror has irradiance of approximately 3 mW/cm².Typically, the first of the 48 holograms is recorded with an energy doseof 35 mJ/cm², and the dose is increased by about 1.5% for eachsubsequent hologram. The total energy dose for recording all 48holograms is typically about 2.5 J/cm². Irradiance and energy dosesdescribed here are merely exemplary. Other embodiments of skew mirrorsand methods of making skew mirrors may use different levels ofirradiance and energy dose.

A first hologram is recorded using a first recording beam internal angle156 of +53.218 degrees and a second recording beam internal angle 157 of+154.234 degrees. The skew axis 164 has a skew angle 164 of +13.726degrees relative to surface normal 122. For each subsequent hologram ofthe grating structure, the first and second recording beam internalangles 156, 157 are typically changed by amounts that are approximatelyequal in magnitude to each other, but having opposite signs. Forexample, for a second hologram, the first recording beam internal angleis changed by +0.091 degree and the second recording beam internal angleis adjusted by −0.091 degree, such that the first recording beaminternal angle 156 becomes +53.309 degrees and the second recording beaminternal angle +154.143 degrees. The magnitudes of changes in recordingbeam internal angles from one hologram to the next hologram varyslightly across the 48 volume holograms (i.e. the change in change inrecording beam internal angles from one hologram to the next varies),from 0.091 degree for changes in recording beam internal angles from thefirst hologram to the second hologram, to 0.084 degree for changes inrecording beam internal angles from the 47^(th) hologram to the 48^(th)hologram. However, for each change of first and second recording beaminternal angles, the magnitude of change is the same and the sign isopposite for each of the first and second beam angles. The first andsecond recording beam internal angles 156, 157 for the last (48^(th))hologram of the first embodiment grating structure 105 are +57.332 and+150.120 degrees, respectively. In some embodiments, the magnitude ofchange of the first recording beam internal angle may differ veryslightly from the magnitude of change of the second recording beaminternal angle in order to compensate for system imprecision, for Snelleffects, for dispersion, or for shrinkage of the grating medium thatresults from recording the holograms.

The first recording beam internal angle 156 ranges from +53.218 to+57.332 degrees (a range of 4.114 degrees) and the second recording beaminternal angle 157 ranges from +154.234 to +150.120 degrees (a range of4.114 degrees). As can be seen in FIG. 4, the first and second recordingbeams 154, 155 are symmetrical about the skew axis 161 such that theinternal angle of the first recording beam relative to the skew axis 166(+38.492 degrees for the first hologram) is equal to 180° minus theinternal angle of the second recording beam relative to the skew axis167 (+141.508 degrees for the first hologram) (180°−+141.508°=38.492degrees). The internal angles of the first and second recording beamsrelative to the skew axis 166, 167 are readily calculated from the firstand second recording beam internal angles 156, 157, respectively, andthe skew angle 164. First and second recording beam internal angles(which are defined as internal angles relative to surface normal of thegrating medium) and internal angles relative to the skew axis of thefirst and second recording beams are listed in Table A-5, appended tothis specification in Appendix A. After recording the 48 volumeholograms, the AK174-200 recording medium is light cured by a processfamiliar to persons skilled in the art.

In a variation of the first method of making a skew mirror, a hologramis created by continuously and synchronously adjusting the first andsecond recording beam internal angles while maintaining the symmetry ofthe first and second recording beams about the skew axis as describedabove. Accordingly, a single hologram is recorded while the firstrecording beam is scanned from a first recording beam internal angle of+53.218 degrees to a first recording beam angle of +57.332 degrees.Simultaneously, the second recording beam is scanned from a secondrecording beam internal angle of +154.234 degrees to +150.120 degrees.The single hologram is thus equivalent to the 48 discrete hologramsrecorded with 48 sets of unique first recording beam and secondrecording beam internal angles, and the total energy dose for recordingthe single hologram is typically about the same (2.5 J/cm²) as for the48 holograms.

A Second Method of Making a Skew Mirror

A second method of making a skew mirror is described below. The skewmirror of the second method is the second embodiment skew mirror 200,which is also illustrated in FIGS. 2A and 2B, and whose physicalproperties are described above.

The second method is identical to the first method except that first andsecond recording beam internal angles are different than with the firstmethod, and the grating medium also differs from the first method. Likethe first embodiment, the second embodiment skew mirror benefits fromhaving reflective properties that allow it to reflect light at asubstantially different wavelength, and in particular a considerablylonger wavelength, than the recording beam wavelength.

The grating structure 205 of the second embodiment skew mirror 200 iscreated by recording 49 volume holograms in the grating medium 210. Eachof the 49 holograms of the second method is recorded at its own uniquefirst recording beam internal angle and its own unique second recordingbeam internal angle. The first recording beam internal angle is aninternal angle of the first recording beam relative to surface normal ofthe grating medium and the second recording beam internal angle is aninternal angle of the second recording beam relative to surface normal.Each of the first and second recording beams for the first embodimentskew mirror has irradiance of approximately 3 mW/cm². Typically, thefirst of the 49 holograms is recorded with an energy dose of 35 mJ/cm²,and the dose is increased by about 1.5% for each subsequent hologram.The total dose for recording all 49 holograms is typically about 2.5J/cm².

According to the second method, a first hologram is recorded using afirst recording beam internal angle of +55.913 degrees and a secondrecording beam internal angle of +153.323 degrees. The skew axis has askew angle of +14.618 degrees relative to surface normal. For eachsubsequent hologram of the grating structure, the first and secondrecording beam internal angles are typically changed by amounts that areapproximately equal in magnitude to each other, but having oppositesigns. For example, for recording a second hologram according to thesecond method, the first recording beam internal angle is changed by+0.095 degree and the second recording beam internal angle is adjustedby −0.095 degree, such that the first recording beam internal anglebecomes +56.008 degrees and the second recording beam internal angle+153.228 degrees. The magnitudes of changes in recording beam internalangles from one hologram to the next hologram typically vary slightlyacross the 49 volume holograms (i.e. the change in change in recordingbeam internal angles from one hologram to the next varies), from amagnitude of 0.095 degree for changes in recording beam internal anglesfrom the first hologram to the second hologram, to a magnitude of 0.087degree for changes in recording beam internal angles from the 48^(th)hologram to the 49^(th) hologram. However, the magnitude of change isthe same for each of the first and second recording beam internalangles, and the sign of the change is opposite for each of the first andsecond recording beam internal angles. The first and second recordingbeam internal angles for the last (49^(th)) hologram of the secondembodiment grating structure are +60.252 and +148.984 degrees,respectively. In some embodiments, the magnitude of change of the firstrecording beam internal angle may differ very slightly from themagnitude of change of the second recording beam internal angle in orderto compensate for factors such as system imprecision, Snell effects,dispersion, or shrinkage of the grating medium that results fromrecording the holograms.

Thus according to the second method the first recording beam internalangle ranges from +55.913 to +60.252 degrees (a range of 4.339 degrees)and the second recording beam internal angle ranges from +153.323 to+148.984 degrees (a range of 4.339 degrees). As with the first method,the first and second recording beams of the second method aresymmetrical about the skew axis such that the internal angle of thefirst recording beam relative to the skew axis (+41.295 degrees for thefirst hologram) is equal to 180° minus the internal angle of the secondrecording beam relative to the skew axis (+138.705 for the firsthologram) (180°−+138.705°=+41.295 degrees). According to the secondmethod, the internal angles of the first and second recording beamsrelative to the skew axis are readily calculated from the first andsecond recording beam internal angles respectively, and the skew angle.For the second method of making a skew mirror, first and secondrecording beam internal angles (which are defined as internal anglesrelative to surface normal of the grating medium) and internal anglesrelative to the skew axis for the first and second recording beams arelisted in Table A-6, appended to this specification in Appendix A. Afterrecording the 49 volume holograms, the AK233-200 recording medium islight cured by a process familiar to persons skilled in the art.

In a variation of the second method of making a skew mirror, a hologramis created by continuously and synchronously adjusting the first andsecond recording beam internal angles while maintaining the symmetry ofthe first and second recording beams about the skew axis as describedabove. Accordingly, a single hologram is recorded while the firstrecording beam is scanned from a first recording beam internal angle of+55.913 degrees to a first recording beam angle of +60.252 degrees.Simultaneously, the second recording beam is scanned from a secondrecording beam internal angle of +153.323 degrees to +148.984 degrees.The single hologram is thus equivalent to the 49 discrete hologramsrecorded with 49 sets of unique first recording beam and secondrecording beam internal angles. The total energy dose for recording thesingle hologram is typically 2.5 J/cm² for the single hologram.

A Multiwavelength Method of Making a Skew Mirror

In a multiwavelength method of making a skew mirror, six volumeholograms are recorded in AK233-200 grating medium, with each of the sixholograms being recorded using its own unique first and second recordingbeam internal angles of incidence. In addition, for each of the sixvolume holograms, wavelengths of the first and second recording beamsare adjusted continuously and synchronously from 403 nm to 408 nm, usinga variable wavelength laser. Wavelengths of the first and secondrecording beams are kept equal to each other while recording each of thesix volume holograms. Total energy dose delivered in recording the sixvolume holograms according to the multiwavelength method is typically,but not necessarily, 2.5 J/cm² for First and second recording beaminternal angles of incidence for the multiwavelength method of making askew mirror are provided below in Table 2. A skew mirror made by themultiwavelength method has the same reflective characteristics of thesecond embodiment skew mirror described above.

TABLE 2 RECORDING BEAM ANGLES FOR A MULTIWAVELENGTH METHOD OF MAKING ASKEW MIRROR First Recording Second Recording Beam Angle of Beam Angle ofHOLOGRAM Incidence* Incidence* 1 56.235° 153.001° 2 57.033° 152.203° 357.813° 151.423° 4 58.568° 150.668° 5 59.303° 149.933° 6 60.018°149.218° internal, relative to grating medium surface normalOther Skew Mirror Embodiments

Embodiments of a skew mirror can be created in a grating mediumcomprising a volumetric dielectric medium, such as a photosensitiverecording medium. Skew mirror embodiments may be formed by constraininga spatial dielectric modulation spectrum as described herein. In anembodiment, dielectric modulation is accomplished holographically byrecording an interference pattern of two or more coherent light beams ina photosensitive recording medium. In other embodiments, dielectricmodulation can be accomplished by other means.

k-Space Formalism for Holography

The k-space formalism is a method for analyzing holographic recordingand diffraction [1]. In k-space, propagating optical waves and hologramsare represented by three dimensional Fourier transforms of theirdistributions in real space. For example, an infinite collimatedmonochromatic reference beam can be represented in real space andk-space by equation (1),

$\begin{matrix}{{{E_{r}\left( \overset{\rightharpoonup}{r} \right)} = {{{A_{r}{\exp\left( {{\mathbb{i}}\;{{\overset{\rightharpoonup}{k}}_{r} \cdot \overset{\rightharpoonup}{r}}} \right)}}\overset{\mspace{20mu}\mathcal{J}\mspace{25mu}}{\rightarrow}{E_{r}\left( \overset{\rightharpoonup}{k} \right)}} = {A_{r}\;{\delta\left( {\overset{\rightharpoonup}{k} - {\overset{\rightharpoonup}{k}}_{r}} \right)}}}},} & (1)\end{matrix}$where E_(r)({right arrow over (r)}) is the optical scalar fielddistribution at all {right arrow over (r)}={x, y, z} 3D spatial vectorlocations, and its transform E_(r)({right arrow over (k)}) is theoptical scalar field distribution at all {right arrow over(k)}={k_(x),k_(y),k_(z)} 3D spatial frequency vectors. A_(r) is thescalar complex amplitude of the field; and {right arrow over (k)}_(r) isthe wave vector, whose length indicates the spatial frequency of thelight waves, and whose direction indicates the direction of propagation.In some embodiments, all beams are composed of light of the samewavelength, so all optical wave vectors must have the same length, i.e.,|{right arrow over (k)}_(r)|=k_(n). Thus, all optical propagationvectors must lie on a sphere of radius k_(n)=2πn₀/λ, where n₀ is theaverage refractive index of the hologram (“bulk index”), and λ is thevacuum wavelength of the light. This construct is known as the k-sphere.In other embodiments, light of multiple wavelengths may be decomposedinto a superposition of wave vectors of differing lengths, lying ondifferent k-spheres.

Another important k-space distribution is that of the hologramsthemselves. Volume holograms usually consist of spatial variations ofthe index of refraction within a grating medium. The index of refractionspatial variations, typically denoted Δn({right arrow over (r)}), can bereferred to as index modulation patterns, the k-space distributions ofwhich are typically denoted Δn({right arrow over (k)}). The indexmodulation pattern created by interference between a first recordingbeam and a second recording beam is typically proportional to thespatial intensity of the recording interference pattern, as shown inequation (2),Δn({right arrow over (r)})∝|E ₁({right arrow over (r)})+E ₂({right arrowover (r)})|² =|E ₁({right arrow over (r)})|² +|E ₂({right arrow over(r)})|² +E ₁*({right arrow over (r)})E ₂({right arrow over (r)})+E₁({right arrow over (r)})E ₂*({right arrow over (r)}),  (2)where E₁({right arrow over (r)}) is the spatial distribution of thesignal first recording beam field and E₂({right arrow over (r)}) is thespatial distribution of the second recording beam field. The unaryoperator * denotes complex conjugation. The final term in equation (2),E₁({right arrow over (r)})E₂*({right arrow over (r)}), maps the incidentsecond recording beam into the diffracted first recording beam. Thus wecan write equation (3),

$\begin{matrix}{{{{E_{1}\left( \overset{\rightharpoonup}{r} \right)}{E_{2}^{*}(r)}}\overset{\mspace{14mu}\mathcal{J}\mspace{20mu}}{\rightarrow}{{E_{1}\left( \overset{\rightharpoonup}{k} \right)} \otimes {E_{2}\left( \overset{\rightharpoonup}{k} \right)}}},} & (3)\end{matrix}$where ⊗ is the 3D cross correlation operator. This is to say, theproduct of one optical field and the complex conjugate of another in thespatial domain becomes a cross correlation of their respective Fouriertransforms in the frequency domain.

FIG. 5A illustrates a real space representation of recording a hologram505 in a grating medium 510 using a second recording beam 515 and afirst recording beam 514. The grating medium typically includes arecording layer configured to record interference patterns as holograms.FIG. 5A omits grating medium components other than the recording layer,such as an additional layer that might serve as a substrate orprotective layer for the recording layer. The second recording beam 515and first recording beam 514 are counter-propagating. Each of the secondrecording beam 515 and first recording beam 514 are typically plane wavebeams having the same wavelength as each other, and the first recordingbeam 514 typically contains no information encoded therein that is notalso present in the second recording beam. Thus the first and secondrecording beams, which can be referred to as signal and reference beams,are typically substantially identical to each other except for angles atwhich they are incident upon the recording medium 510.

FIG. 5B illustrates a k-space representation of the first and secondrecording beams, and the hologram. The hologram illustrated in FIGS. 5Aand 5B is a simple Bragg reflection hologram generated with thecounter-propagating first recording beam 514 and second recording beam515, and recorded in recording medium 510. FIG. 5A shows the secondrecording beam 515 and the first recording beam 514 impinging onopposite sides of the grating medium 510. Optical scalar fielddistributions at all {right arrow over (r)}={x, y, z} 3D spatial vectorlocations for each of the second recording beam 515 and the firstrecording beam 514 can be represented as E₂({right arrow over (r)}) andE₁({right arrow over (r)}), respectively. The recording beams 514, 515form planar interference fringes, which are recorded as a hologram 505within the grating medium 510. The hologram 505 comprises a sinusoidalrefractive index modulation pattern, and can be represented as Δn({rightarrow over (r)}). In a counter-propagating configuration, the recordedplanar interference fringes have a spacing exactly half that of the(internal) wavelength of the light used to record the hologram.

FIG. 5B shows a k-space representation of the situation illustrated inreal space by FIG. 5A. The recording beams are represented in FIG. 5B bypoint-like k-space distributions lying on opposite sides of therecording k-sphere 570. As illustrated in FIG. 5B, the second recordingbeam has a k-space distribution 562, and the first recording beam has ak-space distribution 563. The second recording beam k-space distribution562 can be represented as E₂({right arrow over (k)}) and the firstrecording beam k-space distribution 563 can be represented as E₁({rightarrow over (k)}). Each of the second recording beam k-space distribution562 and the first recording beam k-space distribution 563 are“point-like.” Second recording beam wave vector 564 and first recordingbeam wave vector 565, are shown extending from the origin to the secondrecording beam k-space distribution 562 and first recording beam k-spacedistribution 563, respectively. The second recording beam wave vector564 can be represented as E₂({right arrow over (k)}) and the firstrecording beam wave vector 565 can be represented as E₁({right arrowover (k)}). The hologram itself is represented in FIG. 5B by twoconjugate sideband k-space distributions 568, each of which can berepresented as Δn({right arrow over (k)}) and referred to as a Δn({rightarrow over (r)}) k-space distribution. The two Δn({right arrow over(r)}) k-space distributions 568 have a small, finite size, but are“point-like” in the sense that they are typically several orders ofmagnitude smaller than their distance to the origin, or other featuresof FIG. 5B. For instance, if the thickness of grating medium 510 is 200μm with refractive index 1.5 and the recording beams have a wavelengthof 532 nm, then distributions 568 each resemble a sinc function alongthe k_(z) dimension with size 3.14×10⁴ rad/m null-to-null. However,their distance from the origin is 3.56×10⁷ rad/m, which is more than1000 times as large. Unless specified otherwise, all recited wavelengthsrefer to vacuum wavelengths.

Typically, the hologram constitutes a refractive index distribution thatis real-valued in real space. Locations of the two Δn({right arrow over(k)}) k-space distributions 568 of the hologram may be determinedmathematically from the cross-correlation operations E₂({right arrowover (k)})⊗E₁({right arrow over (k)}) and E₁({right arrow over(k)})⊗E₂({right arrow over (k)}), respectively, or geometrically fromvector differences {right arrow over (K)}_(G+)={right arrow over(k)}₁−{right arrow over (k)}₂ and {right arrow over (K)}_(G−)={rightarrow over (k)}₂−{right arrow over (k)}₁, where {right arrow over(K)}_(G+) and {right arrow over (K)}_(G−) are grating vectors from therespective hologram Δn({right arrow over (k)}) k-space distributions tothe origin (not shown individually). A combined grating vector 569,which can be represented as {right arrow over (K)}_(G), comprising both{right arrow over (K)}_(G+) and {right arrow over (K)}_(G−) gratingvectors, is shown in FIG. 5B as double headed arrow 569 extendingbetween the second recording beam k-space distribution 562 and the firstrecording beam k-space distribution 563. Note that by convention, wavevectors are represented by a lowercase “k,” and grating vectors byuppercase “K.”

Once recorded, the hologram may be illuminated by a probe beam toproduce a diffracted beam. For purposes of the present invention, thediffracted beam can be considered a reflection of the probe beam, whichcan be referred to as an incident light beam. The probe beam and itsreflected beam are angularly bisected by a reflective axis (i.e. theangle of incidence of the probe beam relative to the reflective axis hasthe same magnitude as the angle of reflection of the reflected beamrelative to the reflective axis). The diffraction process can berepresented by a set of mathematical and geometric operations in k-spacesimilar to those of the recording process. In the weak diffractionlimit, the diffracted light distribution of the diffracted beam is givenby equation (4),E _(d)({right arrow over (k)})∝Δn({right arrow over (k)})*E _(p)({rightarrow over (k)})|_(|{right arrow over (k)}|=k) _(n) ,  (4)where E_(d) ({right arrow over (k)}) and E_(p) ({right arrow over (k)})are k-space distributions of the diffracted beam and the probe beam,respectively; and “*” is the 3D convolution operator [1]. The notation“|_(|{right arrow over (k)}|=k) _(n) ” indicates that the precedingexpression is evaluated only where |{right arrow over (k)}|=k_(n), i.e.,where the result lies on the k-sphere. The convolution Δn({right arrowover (k)})*E_(p)({right arrow over (k)}) represents a polarizationdensity distribution, and is proportional to the macroscopic sum of theinhomogeneous electric dipole moments of the grating medium induced bythe probe beam, E_(p)({right arrow over (k)}).

Typically, when the probe beam resembles one of the recording beams usedfor recording, the effect of the convolution is to reverse the crosscorrelation during recording, and the diffracted beam will substantiallyresemble the other recording beam used to record the hologram. When theprobe beam has a different k-space distribution than the recording beamsused for recording, the hologram may produce a diffracted beam that issubstantially different than the beams used to record the hologram. Notealso that while the recording beams are typically mutually coherent, theprobe beam (and diffracted beam) is not so constrained. Amultiwavelength probe beam may be analyzed as a superposition ofsingle-wavelength beams, each obeying Equation (4) with a differentk-sphere radius.

FIGS. 6A and 6B illustrate cases of Bragg-matched and Bragg-mismatchedreconstructions, respectively, generated by illuminating the hologramdepicted in FIGS. 5A and 5B. In both the Bragg-matched andBragg-mismatched cases, the hologram is illuminated with a probe beamhaving a shorter wavelength than the recording beams used to record thehologram. The shorter wavelength corresponds to a longer wave vector.Accordingly, a probe k-sphere 572 has a greater radius than that of therecording k-sphere 570. Both the probe k-sphere 572 and the recordingk-sphere 570 are indicated in FIGS. 6A and 6B.

FIG. 6A shows a case where the probe beam is designed to produce adiffracted beam k-space distribution 575 (represented as E_(d)({rightarrow over (k)})) that is point-like and lies on the probe beam k-sphere572. The diffracted beam k-space distribution 575 is produced accordingto the convolution of Equation (4). The probe beam has a k-spacedistribution 576 (represented as E_(p)({right arrow over (k)})) that isalso point-like. In this case, the probe beam is said to be“Bragg-matched” to the hologram, and the hologram may producesignificant diffraction, even though the probe beam wavelength differsfrom the wavelength of the recording beams used to record the hologram.As best seen in FIG. 6A, the convolution operation may also berepresented geometrically by the vector sum {right arrow over(k)}_(d)={right arrow over (k)}_(p)+{right arrow over (K)}_(G+), where{right arrow over (k)}_(d) represents a diffracted beam wave vector 577,{right arrow over (k)}_(p) represents a probe beam wave vector 578, and{right arrow over (K)}_(G+) represents a sideband grating vector 579.

FIG. 6A shows a k-space representation of a mirror-like diffraction(which can be referred to as a reflection) of the probe beam by thehologram, where the probe beam angle of incidence with respect to thek_(z) axis is equal to the diffracted beam angle of reflection withrespect to the k_(z) axis. FIG. 6B shows a k-space representation of aBragg-mismatched case, wherein a k-space polarization densitydistribution 580, which can be represented as Δn({right arrow over(k)})*E_(p) ({right arrow over (k)}), does not lie on the probe k-sphere572, and thus no significant diffraction of the probe beam occurs. Thisnon-diffracted k-space distribution 580 in the Bragg-mismatched caseillustrated in FIG. 6B is somewhat analogous to the diffracted beamk-space distribution 575 in the Bragg-matched case illustrated in FIG.6A, but k-space distribution 580 should not be referred to as adiffracted beam k-space distribution because no significant diffractionof the probe beam occurs.

Comparing the Bragg-matched and Bragg-mismatched cases, it is evidentthat the hologram will only produce mirror-like diffraction over a verysmall range of input angles for a given probe wavelength, if at all.Those skilled in the art will recognize that this range may be somewhatextended by over-modulating the hologram, or by using a very thinrecording layer; but that these steps may still not lead to mirror-likebehavior over a larger range of wavelengths and angles. These steps mayalso lead to undesired chromatic dispersion.

Skew Mirror Embodiment in k-Space

FIGS. 5A, 5B, 6A, and 6B represent a prior art reflection hologramconstituted by a single sinusoidal grating. As illustrated, thishologram exhibits mirror-like reflectivity in a narrow band ofwavelengths and incidence angles. The specific properties of such ahologram may be determined by application of the well-known coupled wavetheory of Kogelnik [2]. Conversely, embodiments of the present inventionexhibit novel mirror-like reflectivity across relatively broad ranges ofwavelengths and angles by creating a more complex grating structurecomprising multiple gratings.

FIG. 7 shows a geometry illustrating the Bragg selectivity of a singlesinusoidal grating. Grating medium 710 contains a single sinusoidalgrating of thickness d which reflects incident light 724 of a singlewavelength, λ₀, as principal reflected light 727. At the Bragg-matchedcondition, incident light 724 impinges at angle θ_(i), and reflects asreflected light 727 at angle θ_(r), both angles measured with respect tothe z axis. Incident light 724 and reflected light 727 also define areflective axis 738, about which the angular magnitudes of incidenceθ_(i)′ and reflection θ_(r)′ are equal. Reflective axis 738 is thus anangular bisector of incident light 724 and reflected light 727.

As is known to those skilled in the art, the sinusoidal grating of FIG.7 will exhibit both angular and wavelength Bragg selectivity. Ifincident light 724 impinges at non-Bragg-matched angle θ_(i)+Δθ_(i), thediffraction efficiency may be diminished compared to the Bragg-matcheddiffraction efficiency. The selectivity of a sinusoidal grating may becharacterized by its angular Bragg selectivity, Δθ_(B), given byequation (5):

$\begin{matrix}{{\Delta\;\theta_{B}} = {\frac{\lambda\;\cos\;\theta_{r}}{n_{0}\; d\;{\sin\left( {\theta_{i} - \theta_{r}} \right)}}.}} & (5)\end{matrix}$Those skilled in the art will recognize that in a weakly-diffractingsinusoidal grating, the angle θ_(i)+Δθ_(B) represents the first null inthe angular diffraction efficiency plot. The quantity Δθ_(B) can thus besaid to represent the angular width of the sinusoidal grating in thatdiffraction can be greatly diminished when the angle of incidencedeviates from the Bragg-matched angle θ_(i) by more than several timesΔθ_(B). Similarly, for a weakly-diffracting sinusoidal grating, theskilled artisan would expect a reflective axis to vary considerably formonochromatic incident light whose angle of incidence varies by morethan several times Δθ_(B).

Conversely, skew mirrors according to present invention exhibitrelatively stable diffraction and a substantially constant reflectiveaxis angle for incident light whose angle of incidence varies by manytimes Δθ_(B). Some skew mirror embodiments exhibit a substantiallyconstant reflective axis angle across a range of incident light anglesof incidence of 20×Δθ_(B). In embodiments, reflective axis angles acrossa range of incident light angles of incidence of 20×Δθ_(B) change byless than 0.250 degree; or by less than 0.10 degree; or by less than0.025 degree. As shown in Table 3 below, reflective axis angles across arange of incident light angles of incidence of 20×Δθ_(B) for the firstand second embodiment skew mirrors described above change by less than0.020 degree each of the first and second embodiment skew mirrors, atmultiple wavelengths that differ from each other by WF≥0.036.

TABLE 3 CHANGE IN REFLECTIVE AXIS ANGLES ACROSS AN INCIDENCE ANGLE RANGEOF APPROXIMATELY 20 × Δθ_(B) Difference In reflective Skew Mirror AxisIncident Light Embodiment λ* Angles** Angle Range*** Δθ_(B) ^(†) FIRST532 nm 0.012° −3.167° to +0.369° 0.177° EMBODIMENT 513 nm 0.012° −3.111°to +0.313° 0.171° SKEW MIRROR (AK174-200 recording medium) SECOND 532 nm0.019° −7.246° to −4.726° 0.126° EMBODIMENT 513 nm 0.016° −7.202° to−4.770° 0.122° SKEW MIRROR (AK233-200 recording medium) *wavelength ofboth incident and reflected light. **difference in reflective axisangles (internal, relative to surface normal) for incident light havinga change in angle of incidence of approximately 20 × Δθ_(B). ***range ofincident light angles of incidence (internal, relative to surfacenormal) approximately equal to 20 × 4θ_(B), for which the Difference InReflective Axis Angles is reported in this table. ^(†)Δθ_(B) iscalculated for an incident light angle of incidence at the midpoint ofthe Incident Light Angle Range reported in this table.

Similarly, a sinusoidal grating may be characterized by its wavelengthBragg selectivity, Δλ_(B), given by equation (6):

$\begin{matrix}{{\Delta\;\lambda_{B}} = {\frac{\lambda_{0}^{2}\;\cos\;\theta_{r}}{2n_{0}^{2}\; d\;{\sin^{2}\left( {\theta_{i} - \theta_{r}} \right)}}.}} & (6)\end{matrix}$Those skilled in the art will recognize that in a weakly-diffractingsinusoidal grating, the wavelength λ₀+Δλ_(B) represents the first nullin the wavelength diffraction efficiency plot. The quantity Δλ_(B) canthus be said to represent the wavelength width of the sinusoidal gratingin that no significant diffraction will occur when the incidentwavelength deviates from the Bragg-matched wavelength λ₀ by more thanseveral times Δλ_(B). Those skilled in the art will also recognize thatequations (5) and (6) apply to changes in angle and wavelength only,respectively, and that changing both angle and wavelength simultaneouslymay result in another Bragg-matched condition.

A grating may also be characterized by its diffracted angle response.For a sinusoidal grating, the diffracted angle response may be expressedbyΔθ_(r) cos θ_(r)=−Δθ_(i) cos θ_(i).  (7)The diffracted angle response expresses the change in the angle ofreflection, Δθ_(r), in response to small changes in the angle ofincidence, Δθ_(i). In contrast, a true mirror has an angle responseexpressed by equation (8):Δθ_(r)=−Δθ_(i).  (8)A device that has a diffracted angle response substantiallycharacterized by equation (7) may be said to exhibit grating-likereflective behavior, whereas a device that has a diffracted angleresponse substantially characterized by equation (8) may be said toexhibit mirror-like reflective behavior. A device exhibitinggrating-like reflective behavior will necessarily also exhibit areflective axis that changes with angle of incidence, unless thatreflective axis is normal to the device surface, in which case cosθ_(r)=cos θ_(i). Accordingly, requirements for a relatively simpledevice that reflects light about a reflective axis not constrained tosurface normal, and whose angle of reflection for angles of incidencespanning multiples of its angular Bragg selectivity is constant atwavelengths spanning multiples of its wavelength Bragg selectivity, maynot be met by a single sinusoidal grating.

FIG. 7 illustrates a device geometry in a reflective configuration.Those skilled in the art will recognize that the preceding analysis alsoapplies to device geometries in transmissive configurations and todevice geometries in which one or both beams are waveguided by totalinternal reflection within the device.

FIGS. 8A and 8B illustrate operation of a skew mirror in k-spaceaccording to an embodiment. FIG. 8A shows two Δn({right arrow over (k)})k-space distributions 888 for a hologram recorded in a grating mediumand configured to produce multiwavelength mirror-like diffractionaccording to an embodiment. As explained above with respect to a k-spacerepresentation of a prior art hologram shown in FIG. 5B, a Δn({rightarrow over (k)}) k-space distribution can be represented as Δn({rightarrow over (k)}). A red k-sphere 890, green k-sphere 892, and bluek-sphere 893 in FIGS. 8A and 8B indicate k-spheres corresponding towavelengths of light residing in the red, green, and blue regions of thevisible spectrum, respectively.

Instead of two Δn({right arrow over (k)}) k-space distributionsconstituting a single sinusoidal grating (and which therefore can becharacterized as “point-like”), the Δn({right arrow over (k)}) k-spacedistributions 888 shown in FIG. 8A are situated along a substantiallystraight line in k-space, and thus can be characterized as “linesegment-like”. In some embodiments, line segment-like Δn({right arrowover (r)}) k-space distributions comprise continuously-modulatedsub-segments of a substantially straight line in k-space. In someembodiments, line segment-like Δn({right arrow over (k)}) k-spacedistributions substantially consist of point-like distributions situatedalong a substantially straight line in k-space. The line segment-likeΔn({right arrow over (k)}) k-space distributions 888 are situatedsymmetrically about the origin, and thus may be realized as conjugatesidebands of a real-valued refractive index distribution in real space(represented as Δn({right arrow over (r)})). In some embodiments, themodulation may include absorptive and/or emissive components, and thusmay not exhibit conjugate symmetry in k-space. The complex amplitude ofthe distribution may be uniform, or it may vary in amplitude and/orphase while still exhibiting substantially multiwavelength mirror-likediffraction according to embodiments of the present invention. In anembodiment, the line segment-like Δn({right arrow over (k)}) k-spacedistributions are situated substantially along the k_(z) axis, which, byconvention, is the thickness direction of a recording layer.

FIG. 8B illustrates a multiwavelength mirror-like reflective property ofthe hologram. Illumination of the hologram by a collimated probe beamwith point-like k-space distribution 876 (represented as E_(p)({rightarrow over (k)})) results in a k-space polarization density distribution880 (represented as Δn({right arrow over (k)})*E_(p)({right arrow over(k)})) according to Equation (4). Because the probe beam k-spacedistribution 876 is point-like, polarization density distribution 880resembles a simple translation of Δn({right arrow over (k)}) k-spacedistribution 888 from the origin to the tip of probe beam wave vector878 ({right arrow over (k)}_(p)). Then, also according to Equation (4),only the part of the k-space polarization density distribution 880(Δn({right arrow over (k)})*E_(p)({right arrow over (k)})) intersectingthe k-sphere 892 of the probe beam k-space distribution 876(E_(p)({right arrow over (k)})) contributes to diffraction. Thisproduces the diffracted beam k-space distribution 875, (E_(d)({rightarrow over (r)})), constituting the diffracted beam. Because Δn({rightarrow over (k)}) k-space distribution 888 resembles a line segmentparallel to the {right arrow over (k)}_(z) axis, it is evident that themagnitude of the angle of reflection 882 (θ_(r),) is substantially equalto the magnitude of the angle of incidence 881 (θ_(i),) so that thehologram exhibits mirror-like behavior. Furthermore, it is also evidentthat this property typically holds for any incidence angle andwavelength that produces any diffraction at all, and for anysuperposition of probe beams producing diffraction. A k-spacepolarization distribution Δn({right arrow over (r)})*E_(p)({right arrowover (k)}) will intersect the probe k-sphere at a single point withmirror-symmetry about the k_(x) axis (or about the k_(x), k_(y) plane inthe 3D case). Thus, the hologram of FIG. 8A is configured to exhibitmirror-like behavior at a relatively broad range of wavelengths andangles, and thus constitutes a broadband holographic mirror.

Embodiments typically, but not necessarily, exhibit a gap in Δn({rightarrow over (k)}) k-space distribution 888 near the origin, as shown inFIG. 8A. The presence of the gap can limit performance at very high Δθ(i.e., grazing angles of both incidence and reflection).

According to an embodiment, a skew mirror Δn({right arrow over (k)})k-space distribution may be rotated to an arbitrary angle with respectto the k_(x), k_(y), and k_(z) axes. In some embodiments, the Δn({rightarrow over (k)}) k-space distribution is not perpendicular to therelevant reflecting surface in real space. In other words, thereflective axis of a skew mirror embodiment is not constrained tocoincident with surface normal.

FIGS. 9A and 9B illustrate a skew mirror in k-space. FIGS. 9A and 9B areidentical to FIGS. 8A and 8B, respectively, excepting that alldistributions and vectors have been rotated by approximately 45° aboutthe origin. Following the discussion of FIG. 8B, it is evident that theskew mirror of FIG. 9B also produces mirror-like diffraction for allprobe beam wavelengths and angles that produce diffraction. Thediffraction is mirror-like with respect to the reflective axis 861defined by the line segment-like Δn({right arrow over (k)}) k-spacedistribution 888, i.e., the angle of incidence 881 magnitude withrespect to the reflective axis 861 is equal to the angle of reflection882 magnitude with respect to the reflective axis 861. FIG. 9Billustrates one such case.

FIG. 10A illustrates the operation of a skew mirror in real space. Skewmirror 1010 is characterized by reflective axis 1038 at angle −13°measured with respect to the z axis, which is normal to the skew mirrorsurface 1012. Skew mirror 1010 is illuminated with incident light 1024with internal incidence angle −26° measured with respect to the z axis.Principal reflected light 1027 is reflected with internal reflectionangle 180° measured with respect to the z axis.

FIG. 10B illustrates the skew mirror 1010 of FIG. 10A in k-space. Linesegment-like Δn({right arrow over (k)}) k-space distribution 1088 passesthrough the origin, and has an angle of −13° with respect to the z axis,equal to that of reflective axis 1038. Recording k-sphere 1070 is thek-sphere corresponding to the writing wavelength of 405 nm. A redk-sphere 1090, green k-sphere 1092, and blue k-sphere 1093 in FIGS. 10Band 10D indicate k-spheres corresponding to wavelengths of lightresiding in the red, green, and blue regions of the visible spectrum,respectively.

FIG. 10C illustrates a highly magnified portion of FIG. 10B showing theleft intersection between recording k-sphere 1070 and line segment-likeΔn({right arrow over (k)}) k-space distribution 1088 according to anembodiment. In this view, line segment-like Δn({right arrow over (k)})k-space distribution 1088 can be seen to be include multiple discreteholograms. Each of the multiple discreet holograms 1005 is representedby a horizontal line demarking the first null-to-first null spacing ofthe hologram in the k_(z) direction. In some embodiments, the spacing ofthe discrete holograms may be higher or lower than illustrated in 10C.In some embodiments, the spacing may be low enough to create gaps inline segment-like Δn({right arrow over (k)}) k-space distribution 1088.In some embodiments with gaps, the use of broadband illumination maysubstantially mask any effect of the gaps upon the reflected light. Insome embodiments, this approach may result in a net diffractionefficiency increase. In other embodiments, the spacing of the discreteholograms may be so dense as to approximate or be equivalent to acontinuous distribution.

FIG. 10D illustrates the reflection of blue incident light by the skewmirror of FIG. 10A in k-space. Incident light having a probe beam wavevector 1078 impinges with an internal incidence angle of −26° measuredwith respect to the z axis. The tip of probe beam wave vector 1078 lieson blue k-sphere 1093, indicating the position of point-like probe beamk-space distribution 1076 (E_(p)({right arrow over (k)})). Polarizationdensity distribution 1080 is given by the convolution Δn({right arrowover (k)})*E_(p)({right arrow over (k)}), which resembles linesegment-like Δn({right arrow over (k)}) k-space distribution 1088 (seenin FIG. 10C) translated to the tip of probe beam wave vector 1078.Principal reflected light having diffracted beam wave vector 1077 isdetermined from equation (4) by evaluating polarization densitydistribution 1080 at blue k-sphere 1093. Principal reflected lighthaving diffracted beam wave vector 1077 is reflected with internalpropagation angle 180° measured with respect to the z axis.

Persons skilled in the art will recognize that the term probe beam,typically used here when describing skew mirror properties in k-space,is analogous to the term incident light, which is typically used herewhen describing skew mirror reflective properties in real space.Similarly, the term diffracted beam, typically used here when describingskew mirror properties in k-space, is analogous to the term principalreflected light, typically used here when describing skew mirrorproperties in real space. Thus when describing reflective properties ofa skew mirror in real space, it is typical to state that incident lightis reflected by a hologram (or other grating structure) as principalreflected light, though to state that a probe beam is diffracted by thehologram to produce a diffracted beam says essentially the same thing.Similarly, when describing reflective properties of a skew mirror ink-space, it is typical to state that a probe beam is diffracted by ahologram (or other grating structure) to produce a diffracted beam,though to state that incident light is reflected by the gratingstructure to produce principal reflected light has the same meaning inthe context of embodiments of the present invention.

As shown in FIG. 10D, probe beam wave vector 1078 and diffracted beamwave vector 1077 necessarily form the legs of a substantially isoscelestriangle with line segment-like polarization density distribution 1080as the base. The equal angles of this triangle are necessarily congruentwith the angle of incidence, 1008, and angle of reflection 1009, bothmeasured with respect to reflective axis 1038. Thus, skew mirror 1010reflects light in a substantially mirror-like manner about reflectiveaxis 1038.

The isosceles triangle construction of FIG. 10D obtains wheneverΔn({right arrow over (k)}) k-space distribution 1088 substantiallyresembles a segment of a line passing through the origin, as shown inFIG. 10C. Polarization density distribution 1080 hence substantiallyresembles the straight base of an isosceles triangle, leading tomirror-like reflection about reflective axis 1038 for any incidentinternal wave vectors of any length that diffracts. In some embodiments,dispersion of the grating medium may cause internal wave vectors of thesame direction but differing lengths to refract in different directionsin an external medium according to Snell's law. Similarly, dispersionmay cause external wave vectors of the same direction and differinglengths to refract in different directions in the internal gratingmedium. Accordingly, if it is desired to minimize the effects ofdispersion in a skew mirror, it may be desirable to impart a curve toline segment-like Δn({right arrow over (k)}) k-space distribution 1088,or to otherwise deviate from a line that passes through the origin. Suchan approach may reduce net angular dispersion in reflections involvingexternal refraction according to some metric. Since the dispersion ofuseful grating media is typically quite low, the required deviation froma straight line passing through the origin is small. Adjustments to linesegment-like Δn({right arrow over (k)}) k-space distribution 1088 thatcompensate for dispersion do not fall outside the scope of the presentinvention.

FIG. 11A illustrates the reflection of green incident light by the skewmirror of FIG. 10A in k-space. Incident light with wave vector 1178Aimpinges with internal propagation angle −35° measured with respect tothe z axis. Principal reflected light with wave vector 1177A isreflected with internal propagation angle −171° measured with respect tothe z axis. The magnitudes of angle of incidence 1108A and angle ofreflection 1109A are both substantially equal to 22 degrees measuredwith respect to reflective axis 1038, thus constituting a mirror-likereflection about reflective axis 1038.

FIG. 11B illustrates the reflection of red incident light by the skewmirror of FIG. 10A in k-space. Incident light having probe beam wavevector 1178B impinges with internal propagation angle −35° measured withrespect to the z axis. Principal reflected light having diffracted beamwave vector 1177B is reflected with internal propagation angle −171°measured with respect to the z axis. The magnitudes of angle ofincidence 1108B and angle of reflection 1109B are both substantiallyequal to 22° measured with respect to reflective axis 1038, thusconstituting a mirror-like reflection about reflective axis 1038.

FIGS. 11A and 11B show the reflection of green and red light at the sameangles of incidence and reflection, illustrating the achromaticreflection property of the skew mirror. Those skilled in the art willrecognize that the geometrical constructions of FIGS. 10A-D and 11A-Bwill produce mirror-like reflection at all angle/wavelength combinationsthat produce reflection, including angles and wavelengths notspecifically illustrated.

Skew Mirror Optical Properties

Embodiments of a skew mirror effect a mirror-like reflection withrespect to internal propagation angles, external angles must bedetermined using Snell's law at the relevant boundaries. Because ofthis, a skew mirror may introduce aberrations, dispersion, and/or fielddistortion to external wavefronts. In some embodiments, aberrations,dispersion, and/or field distortions may be mitigated by the use ofcompensating optics. In some embodiments, the compensating optics mayinclude another skew mirror in a symmetric relationship.

A relatively thin skew mirror may introduce lowered angular resolutionin the reflected beam in proportion to the beam's projection onto thethin axis. In some cases it may be advantageous to increase thethickness of the recording layer in order to mitigate this effect.

Skew Mirror Reflectivity

Embodiments of a skew mirror may be either fully or partiallyreflective. Embodiments of a skew mirror may require relatively highdynamic range recording medium to achieve high reflectivity over arelatively wide wavelength bandwidth and angle range. In an embodiment,a skew mirror with an angular range spanning 105° at 405 nm down to 200at 650 nm may require 183 individual holograms in a 200 μm recordinglayer. This configuration has a reflectivity of approximately 7.5% usinga state-of-the-art photosensitive recording medium with a maximumrefractive index modulation of 0.03 [3]. In some embodiments, increasingrecording medium thickness may not lead to increased reflectivity sincediffractive selectivity also increases with thickness.

Skew Mirror Applications

The preceding exposition pertains to internal wavelengths andpropagation angles, although in one case a slab-like hologram withthickness in the z direction was described. Many other configurationsare possible within the scope of the invention. Without implyinglimitation, a few exemplary embodiments are illustrated here.

FIG. 12A illustrates an embodiment referred to as a skew windowcomprising grating structure 1205 in a grating medium, and including areflective axis 1261 about which incident light is symmetricallyrefracted. The skew window is a transmissive analog of the skew mirror.FIG. 12B shows a skew coupler embodiment, which uses a skew mirror tocouple external light into or out of a waveguide 1294. Transmissive skewcouplers are also possible. FIG. 12C shows a skew prism embodiment,which may fold an optical path and/or invert an image.

FIG. 13A illustrates a pupil relay embodiment formed by a slab waveguide1394 with two skew couplers, each of which comprises a grating medium1310 having a reflective axis 1361 that differs from surface normal ofthe grating medium. Since this device is configured to relay input raysto output rays with a uniform 1:1 mapping, it can transmit an image atinfinity through the waveguide 1394 to the eye or other sensor. Such aconfiguration may be useful for head mounted displays (HMDs), amongother applications. In the reverse direction, it may relay an image ofthe eye, possibly for the purposes of eye tracking. FIG. 13B shows askew mirror 1300 used as a concentrator/diffuser, which can transform alarge dim beam into a bright small one, and/or vice-versa.

FIGS. 14A and 14B illustrate an angle filter embodiment of a skewmirror. In FIG. 14A, a Δn({right arrow over (k)}) k-space 1488distribution is indicated with a higher low frequency cut-off (i.e.,larger center gap) compared to the distribution illustrated in FIG. 8A.As a consequence, the skew mirror will reflect only the low θ (i.e.,near normal incidence) angular components of narrow band incident beamE_(inc), into reflected beam E_(r), while transmitting high θ angularcomponents in E_(t). One skilled in the art will readily discern that anarbitrary circularly-symmetric transfer function may be so realized bymodulating the amplitude and/or phase of the line segment-like Δn({rightarrow over (k)}) distribution according to an embodiment of theinvention. Angular filtering may also be accomplished with skew mirrors,and in configurations involving multiple skew mirrors recorded in one ormore media. These configurations may not be constrained to becircularly-symmetric, and may achieve some level of achromaticoperation.

FIG. 15 illustrates another skew mirror embodiment, a “narcissist'smirror” includes several skew mirrors 1500 whose reflective axes 1561intersect. A narcissist can sit at the point of convergence and seeseveral images of them self.

Skew Mirror Fabrication

Skew mirrors may be recorded holographically according to an embodiment.Skew mirrors may be recorded holographically or fabricated by withnon-holographic means according to embodiments.

Holographic Recording

FIGS. 16A and 16B illustrate additional methods for recording skewmirrors. In FIG. 16A, substantially collimated recording beams are usedto illuminate a grating medium to create a desired Δn({right arrow over(k)}) distribution. In one embodiment, illustrated in FIG. 16A, arecording beam set consisting of a first recording beam 1654A and asecond recording beam 1655A at wavelength λ illuminate the gratingmedium 1610 in order to record a first point-like subset of the desiredline segment-like Δn({right arrow over (k)}) distribution, e.g. thehighest-frequency components (the outer tips of Δn({right arrow over(k)})). The angles of incidence θ₁ and θ₂ of a recording apparatus arethen adjusted to produce another set of recording beams consisting ofanother first recording beam 1654B and another second recording beam1655B, which are also at wavelength λ. The other first and secondrecording beams 1654B, 1655B illuminate the medium to record a secondpoint-like subset of the desired line segment-like Δn({right arrow over(k)}) distribution. This process is repeated using yet another set ofrecording beams consisting of yet another first recording beam 1654C andyet another second recording beam 1655C etc. . . . , until an entiredesired line segment-like Δn({right arrow over (k)}) distribution hasbeen recorded.

In some embodiments, this recording may be made in one continuousexposure wherein θ_(r) and θ_(s) are adjusted continuously andsynchronously in order to produce the desired distribution. In otherembodiments, separate, discreet exposures where θ_(r) and θ_(s) arefixed during exposure and changed only between exposures are used. Stillother embodiments may combine these methods. In some embodiments, thecomponents of Δn({right arrow over (k)}) may be written in an arbitraryorder. In some embodiments, intensity may be varied across one or bothbeams in order to control the spatial diffraction efficiency profile. Insome embodiments, a phase control element (e.g., a mirror mounted on apiezo-electric actuator) may be inserted into one or both beam paths inorder to control the phase of each exposure. In some embodiments, morethan one skew mirror or broadband skew mirror might be recorded into thesame medium.

In the case of discreet exposures, the number and angular density ofexposures is sufficient to produce a smooth, continuous linesegment-like Δn({right arrow over (k)}) distribution. One skilled in theart will readily calculate the angular selectivity of each hologramproduced by a discreet exposure using Kogelnik's theory [2]. In oneembodiment, exposures are made at angular increments corresponding to afunction of this angular selectivity, e.g., at the angular spacing ofthe full-width-quarter-maximum (FWQM) of the diffraction efficiencypeaks. In other embodiments, the angular exposure density might be finerthan this in order to assure a smooth final distribution.

The number of FWQM peaks necessary to span the line segment-likeΔn({right arrow over (k)}) distribution may be regarded as an equivalentnumber of holograms, M, required to form the distribution. Accordingly,the maximum possible diffraction efficiency of the resulting skew mirrormay be estimated by η=(M/M/#)² where η is the diffraction efficiency,and M/# is a material parameter characterizing the dynamic range of therecording medium [4]. One skilled in the art will readily determine howto refine this estimate according to the geometry of each individualexposure or the overlap of neighboring exposures.

FIG. 16B illustrates an embodiment where a first prism 1659A and asecond prism 1659B are incorporated to produce internal beam angles thatare not otherwise accessible due to refraction at the grating medium1610 surface. This method is typically used, for example, to fabricatethe skew coupler of FIG. 12B. One skilled in the art will readilyperceive how to modify the configurations of FIGS. 13A and 13B toachieve a desired distribution.

In some embodiments, a single recording wavelength λmay be chosen towrite the entire line segment-like Δn({right arrow over (k)})distribution. For example, in an embodiment it is possible to write askew mirror that operates across all visible wavelengths using only a405 nm laser source. This has an advantage of requiring sufficientrecording medium sensitivity at only a single wavelength, as well as anadvantage of simplicity. In some embodiments, more than one recordingwavelength is used. In still other cases, a continuously-variablewavelength source is used. In one such embodiment, the recording anglesθ_(r) and θ_(s) are held constant, and the recording wavelength isinstead changed in order to produce the entire line segment-likeΔn({right arrow over (k)}) distribution, or a subset thereof.

Other Fabrication Methods

Other methods for producing a skew mirror fall within the scope of thepresent invention. In one embodiment, for example, a very thickdielectric layer structure is built up using conventional opticalcoating means. The structure is designed to produce broadbandreflectivity within sub-layers, typically by repetition of aconventional broadband reflective coating design. The thick structure isthen ground and polished to produce a surface at an oblique angle to thecoating layers. The resulting structure typically exhibits mirror-likebehavior with respect to a reflective axis substantially defined by thenormal of the coating layers rather than the polished surface, and thusconstitutes a skew mirror. In some embodiments, atomically-precisemanufacturing methods enable fabrication of skew mirrors by composingdielectric structures atom-by-atom without regard to external surfaces.

Non-Flat Mirrors

Skew mirrors may be said to be non-flat in two senses: 1) When thephysical shape of the recording medium is not flat; and 2) when theholographic fringes are not planar.

Non-Slab-Like Mirrors

Embodiments of mirrors according to the present invention, includingexamples of skew mirrors, broadband mirrors, and holographic mirrors,include holograms recorded in medium that is not slab-like in shape. Inan example, in an embodiment, a recording layer is cast with a uniformthickness, but on a curved surface. In another example, a non-uniformrecording layer (e.g., wedge-shaped) is utilized. In still anotherexample, an arbitrary shape (e.g., spherical) is molded. In thesenon-slab-like mirror cases, whether the designation “skew mirror” isappropriate depends on the geometry of the relevant surface(s).Non-slab-like holographic mirrors typically exhibit broadbandmirror-like properties.

Mirrors with Non-Planar Holographic Fringes

In some embodiments, it is desirable to introduce optical power or otherdeliberate aberrations into a reflection. This can be accomplished withan embodiment of a skew mirror by locally varying the direction of thereflective axis, for example so that a plane-wave incident beam isreflected to form a spherical-wave reflected beam, as occurs with aconventional parabolic mirror. Such a skew mirror can be fabricated, forinstance, by using one converging and one diverging beam in thefabrication method of FIG. 13 and by recording while changing thewavelength instead of the angle of incidence. Such a mirror can also befabricated by polishing dielectric layers deposited on a non-flatsurface, or by using advanced atomically-precise manufacturing methods.

Other Fabrication Embodiments

Some holographic recording system embodiments incorporates mirrors,lenses and prisms to direct first and second recording beams into thegrating medium in such a way that translation of the grating medium isnot required to record multiple holograms at varying recording beaminternal angles, at approximately the same location in the gratingmedium.

In some embodiments a prism in addition to the coupling prism may beused to fabricate the skew mirror. In some embodiments a variety ofcoupling prisms and flat pieces of glass may be used. In someembodiments multiple beams, E_(r) _(_) _(N) and E_(s) _(_) _(N), atmultiple wavelengths, λ_(N), may be. In some embodiments multiplewavelengths may be used to fabricate multiple discrete line segment-likeΔn({right arrow over (k)}) distributions. In some embodiments multiplewavelengths may be used to fabricate a line segment-like Δn({right arrowover (k)}) distribution that may be continuous or may include closelyspaced sections. In some embodiments the incident angle of the signaland/or reference beam may be adjusted to compensate for shrinkage of thesample material. In some embodiments the sample may be rotated tocompensate for shrinkage of the sample material. In some embodiments thewavelength may be changed to compensate for shrinkage of the samplematerial.

Alternative Embodiments and Variations

The various embodiments and variations thereof, illustrated in theaccompanying Figures and/or described above, are merely exemplary andare not meant to limit the scope of the invention. It is to beappreciated that numerous other variations of the invention have beencontemplated, as would be obvious to one of ordinary skill in the art,given the benefit of this disclosure. All variations of the inventionthat read upon appended claims are intended and contemplated to bewithin the scope of the invention.

Terminology

The terms and phrases as indicated in quotation marks (“ ”) in thissection are intended to have the meaning ascribed to them in thisTerminology section applied to them throughout this document, includingin the claims, unless clearly indicated otherwise in context. Further,as applicable, the stated definitions are to apply, regardless of theword or phrase's case, to the singular and plural variations of thedefined word or phrase.

References in the specification to “one embodiment,” “an embodiment,”“another embodiment,” “a preferred embodiment,” “an alternativeembodiment,” “one variation,” “a variation,” and similar phrases meanthat a particular feature, structure, or characteristic described inconnection with the embodiment or variation, is included in at least anembodiment or variation of the invention. The phrase “in oneembodiment,” “in one variation” or similar phrases, as used in variousplaces in the specification, are not necessarily meant to refer to thesame embodiment or the same variation.

The term “approximately,” as used in this specification and appendedclaims, refers to plus or minus 10% of the value given.

The term “about,” as used in this specification and appended claims,refers to plus or minus 20% of the value given.

The term “generally,” as used in this specification and appended claims,mean mostly, or for the most part.

The term “principally,” as used in this specification and appendedclaims with respect to reflected light, refers to light reflected by agrating structure. Light that is principally reflected at a recitedangle includes more light than is reflected at any other angle(excluding surface reflections). Light that is principally reflectedabout a recited reflective axis includes more reflected light than isreflected about any other reflective axis (excluding surfacereflections). Light reflected by a device surface is not included whenconsidering principally reflected light.

The term “reflective axis”, as used in this specification and appendedclaims, refers to an axis that bisects an angle of incident lightrelative to its reflection. The absolute value of an angle of incidenceof the incident light relative to the reflective axis is equal to theabsolute value of the angle of reflection of the incident light'sreflection, relative to the reflective axis. For prior art dielectricmirrors, the reflective axis is coincident with surface normal, i.e. thereflective axis is perpendicular to the mirror surface. Conversely,embodiments of skew mirrors according to the present invention may havea reflective axis that differs from surface normal, or may have areflective axis that is coincident with surface normal. Persons skilledin the art will recognize that a reflective axis angle can determined byadding an angle of incidence to its respective angle of reflection, anddividing the resulting sum by two. Angles of incidence and angles ofreflection are typically determined empirically, with multiplemeasurements (generally three or more) typically used to generate a meanvalue.

The term “angle interval” and “angle intervals,” as used in thisspecification and appended claims, refers to regular, angular spacingbetween multiple light beams incident upon a skew mirror across a rangeof angles of incidence. Accordingly, each of the multiple incident lightbeams has an angle of incidence that differs from angles of incidence ofall others of the multiple incident light beams by a positive integermultiple of the angle interval, and the each of the multiple incidentlight beams has an angle of incidence that differs from the angle ofincidence of at least one other of the multiple incident light beams bythe angle interval.

The term “light,” as used in this specification and appended claims,refers to electromagnetic radiation familiar to persons skilled in theart. Unless reference is made to a specific wavelength or range ofwavelengths, such as “visible light”, which refers to a part of theelectromagnetic spectrum visible to the human eye, the electromagneticradiation can have any wavelength.

The terms “hologram” and “holographic grating,” as used in thisspecification and appended claims, refers to a recording of aninterference pattern generated by interference between multipleintersecting light beams. A hologram or holographic grating is anexample of a grating structure.

While various inventive embodiments have been described and illustratedherein, those of ordinary skill in the art will readily envision avariety of other means and/or structures for performing the functionand/or obtaining the results and/or one or more of the advantagesdescribed herein, and each of such variations and/or modifications isdeemed to be within the scope of the inventive embodiments describedherein. More generally, those skilled in the art will readily appreciatethat all parameters, dimensions, materials, and configurations describedherein are meant to be exemplary and that the actual parameters,dimensions, materials, and/or configurations will depend upon thespecific application or applications for which the inventive teachingsis/are used. Those skilled in the art will recognize, or be able toascertain using no more than routine experimentation, many equivalentsto the specific inventive embodiments described herein. It is,therefore, to be understood that the foregoing embodiments are presentedby way of example only and that, within the scope of the appended claimsand equivalents thereto, inventive embodiments may be practicedotherwise than as specifically described and claimed. Inventiveembodiments of the present disclosure are directed to each individualfeature, system, article, material, kit, and/or method described herein.In addition, any combination of two or more such features, systems,articles, materials, kits, and/or methods, if such features, systems,articles, materials, kits, and/or methods are not mutually inconsistent,is included within the inventive scope of the present disclosure.

The above-described embodiments can be implemented in any of numerousways. For example, embodiments of designing and making the technologydisclosed herein may be implemented using hardware, software or acombination thereof. When implemented in software, the software code canbe executed on any suitable processor or collection of processors,whether provided in a single computer or distributed among multiplecomputers.

Further, it should be appreciated that a computer may be embodied in anyof a number of forms, such as a rack-mounted computer, a desktopcomputer, a laptop computer, or a tablet computer. Additionally, acomputer may be embedded in a device not generally regarded as acomputer but with suitable processing capabilities, including a PersonalDigital Assistant (PDA), a smart phone or any other suitable portable orfixed electronic device.

Also, a computer may have one or more input and output devices. Thesedevices can be used, among other things, to present a user interface.Examples of output devices that can be used to provide a user interfaceinclude printers or display screens for visual presentation of outputand speakers or other sound generating devices for audible presentationof output. Examples of input devices that can be used for a userinterface include keyboards, and pointing devices, such as mice, touchpads, and digitizing tablets. As another example, a computer may receiveinput information through speech recognition or in other audible format.

Such computers may be interconnected by one or more networks in anysuitable form, including a local area network or a wide area network,such as an enterprise network, and intelligent network (IN) or theInternet. Such networks may be based on any suitable technology and mayoperate according to any suitable protocol and may include wirelessnetworks, wired networks or fiber optic networks.

The various methods or processes (e.g., of designing and making thecoupling structures and diffractive optical elements disclosed above)outlined herein may be coded as software that is executable on one ormore processors that employ any one of a variety of operating systems orplatforms. Additionally, such software may be written using any of anumber of suitable programming languages and/or programming or scriptingtools, and also may be compiled as executable machine language code orintermediate code that is executed on a framework or virtual machine.

In this respect, various inventive concepts may be embodied as acomputer readable storage medium (or multiple computer readable storagemedia) (e.g., a computer memory, one or more floppy discs, compactdiscs, optical discs, magnetic tapes, flash memories, circuitconfigurations in Field Programmable Gate Arrays or other semiconductordevices, or other non-transitory medium or tangible computer storagemedium) encoded with one or more programs that, when executed on one ormore computers or other processors, perform methods that implement thevarious embodiments of the invention discussed above. The computerreadable medium or media can be transportable, such that the program orprograms stored thereon can be loaded onto one or more differentcomputers or other processors to implement various aspects of thepresent invention as discussed above.

The terms “program” or “software” are used herein in a generic sense torefer to any type of computer code or set of computer-executableinstructions that can be employed to program a computer or otherprocessor to implement various aspects of embodiments as discussedabove. Additionally, it should be appreciated that according to oneaspect, one or more computer programs that when executed perform methodsof the present invention need not reside on a single computer orprocessor, but may be distributed in a modular fashion amongst a numberof different computers or processors to implement various aspects of thepresent invention.

Computer-executable instructions may be in many forms, such as programmodules, executed by one or more computers or other devices. Generally,program modules include routines, programs, objects, components, datastructures, etc. that perform particular tasks or implement particularabstract data types. Typically the functionality of the program modulesmay be combined or distributed as desired in various embodiments.

Also, data structures may be stored in computer-readable media in anysuitable form. For simplicity of illustration, data structures may beshown to have fields that are related through location in the datastructure. Such relationships may likewise be achieved by assigningstorage for the fields with locations in a computer-readable medium thatconvey relationship between the fields. However, any suitable mechanismmay be used to establish a relationship between information in fields ofa data structure, including through the use of pointers, tags or othermechanisms that establish relationship between data elements.

Also, various inventive concepts may be embodied as one or more methods,of which an example has been provided. The acts performed as part of themethod may be ordered in any suitable way. Accordingly, embodiments maybe constructed in which acts are performed in an order different thanillustrated, which may include performing some acts simultaneously, eventhough shown as sequential acts in illustrative embodiments.

All definitions, as defined and used herein, should be understood tocontrol over dictionary definitions, definitions in documentsincorporated by reference, and/or ordinary meanings of the definedterms.

The indefinite articles “a” and “an,” as used herein in thespecification and in the claims, unless clearly indicated to thecontrary, should be understood to mean “at least one.”

The phrase “and/or,” as used herein in the specification and in theclaims, should be understood to mean “either or both” of the elements soconjoined, i.e., elements that are conjunctively present in some casesand disjunctively present in other cases. Multiple elements listed with“and/or” should be construed in the same fashion, i.e., “one or more” ofthe elements so conjoined. Other elements may optionally be presentother than the elements specifically identified by the “and/or” clause,whether related or unrelated to those elements specifically identified.Thus, as a non-limiting example, a reference to “A and/or B”, when usedin conjunction with open-ended language such as “comprising” can refer,in one embodiment, to A only (optionally including elements other thanB); in another embodiment, to B only (optionally including elementsother than A); in yet another embodiment, to both A and B (optionallyincluding other elements); etc.

As used herein in the specification and in the claims, “or” should beunderstood to have the same meaning as “and/or” as defined above. Forexample, when separating items in a list, “or” or “and/or” shall beinterpreted as being inclusive, i.e., the inclusion of at least one, butalso including more than one, of a number or list of elements, and,optionally, additional unlisted items. Only terms clearly indicated tothe contrary, such as “only one of” or “exactly one of,” or, when usedin the claims, “consisting of,” will refer to the inclusion of exactlyone element of a number or list of elements. In general, the term “or”as used herein shall only be interpreted as indicating exclusivealternatives (i.e. “one or the other but not both”) when preceded byterms of exclusivity, such as “either,” “one of,” “only one of,” or“exactly one of.” “Consisting essentially of,” when used in the claims,shall have its ordinary meaning as used in the field of patent law.

As used herein in the specification and in the claims, the phrase “atleast one,” in reference to a list of one or more elements, should beunderstood to mean at least one element selected from any one or more ofthe elements in the list of elements, but not necessarily including atleast one of each and every element specifically listed within the listof elements and not excluding any combinations of elements in the listof elements. This definition also allows that elements may optionally bepresent other than the elements specifically identified within the listof elements to which the phrase “at least one” refers, whether relatedor unrelated to those elements specifically identified. Thus, as anon-limiting example, “at least one of A and B” (or, equivalently, “atleast one of A or B,” or, equivalently “at least one of A and/or B”) canrefer, in one embodiment, to at least one, optionally including morethan one, A, with no B present (and optionally including elements otherthan B); in another embodiment, to at least one, optionally includingmore than one, B, with no A present (and optionally including elementsother than A); in yet another embodiment, to at least one, optionallyincluding more than one, A, and at least one, optionally including morethan one, B (and optionally including other elements); etc.

In the claims, as well as in the specification above, all transitionalphrases such as “comprising,” “including,” “carrying,” “having,”“containing,” “involving,” “holding,” “composed of,” and the like are tobe understood to be open-ended, i.e., to mean including but not limitedto. Only the transitional phrases “consisting of” and “consistingessentially of” shall be closed or semi-closed transitional phrases,respectively, as set forth in the United States Patent Office Manual ofPatent Examining Procedures, Section 221.03.

-   [1] M. R. Ayres, “k-Space Formalism,” in K. Curtis, L. Dhar, W. L.    Wilson, A. Hill, M. R. Ayres, Holographic Data Storage: From Theory    to Practical Systems, John Wiley & Sons, Ltd. (2010), pp. 26-31.-   [2] H. Kogelnik, “Coupled wave theory for thick hologram gratings,”    Bell Sys. Tech. J. 48, 2909-2947 (1969).-   [3] F. R. Askham, “Photopolymer media with enhanced dynamic range,”    U.S. Pat. No. 8,658,332, Feb. 24, 2014.-   [4] F. H. Mok, G. W. Burr, D. Psaltis, “System metric for    holographic memory systems,” Opt. Lett. 21, 896-898 (1996).

APPENDIX A

TABLE A-1 ANGLES OF FIRST INCIDENT LIGHT, FIRST REFLECTED LIGHT, ANDFIRST REFLECTIVE AXIS, FOR A FIRST EMBODIMENT SKEW MIRROR; WAVELENGTH =532 NM; AK174-200 RECORDING MEDIUM Angle Of Angle Of Angle of Angle ofIncidence Reflection Incidence of Reflection of First First First Firstof First of First First First Reflective Internal Reflective InternalIncident Reflected Incident Reflected Axis Angle of Axis Angle Angle ofLight Light Light Light Angle Reflection (INTERNAL, Incidence (INTERNAL,(INTERNAL, (EXTERNAL, (EXTERNAL, (EXTERNAL, (relative relative to(relative relative to relative to relative to relative to relative tosurface surface to surface reflective reflective surface surface tosurface normal, normal, normal, axis, in axis, in normal, in normal, innormal, Measurement in degrees) indegrees) in degrees) degrees) degrees)degrees) degrees) in degrees) 1 25.668 13.800 1.933 −11.867 11.867 2.90040.521 21.711 2 25.680 13.773 1.866 −11.907 11.907 2.800 40.542 21.671 325.691 13.746 1.800 −11.946 11.946 2.701 40.563 21.632 4 25.814 13.7741.733 −12.041 12.041 2.600 40.782 21.691 5 25.938 13.803 1.667 −12.13612.136 2.501 41.003 21.752 6 26.005 13.802 1.600 −12.202 12.202 2.40041.122 21.761 7 25.904 13.719 1.533 −12.185 12.185 2.300 40.942 21.621 825.971 13.719 1.466 −12.252 12.252 2.200 41.062 21.631 9 26.094 13.7471.400 −12.347 12.347 2.101 41.283 21.692 10 26.216 13.775 1.333 −12.44212.442 2.000 41.502 21.751 11 26.339 13.803 1.267 −12.536 12.536 1.90141.723 21.812 12 26.350 13.775 1.200 −12.575 12.575 1.800 41.742 21.77113 26.472 13.803 1.134 −12.669 12.669 1.701 41.963 21.832 14 26.53813.802 1.067 −12.736 12.736 1.600 42.082 21.841 15 26.660 13.830 1.001−12.830 12.830 1.501 42.303 21.902 16 26.780 13.857 0.933 −12.924 12.9241.399 42.521 21.960 17 26.738 13.802 0.867 −12.935 12.935 1.301 42.44321.872 18 26.803 13.801 0.800 −13.001 13.001 1.200 42.561 21.881 1926.923 13.829 0.734 −13.095 13.095 1.101 42.781 21.941 20 26.989 13.8280.667 −13.161 13.161 1.000 42.901 21.951 21 26.946 13.773 0.601 −13.17313.173 0.901 42.822 21.862 22 27.066 13.800 0.533 −13.266 13.266 0.80043.041 21.921 23 26.913 13.690 0.467 −13.223 13.223 0.701 42.762 21.73224 27.088 13.744 0.400 −13.344 13.344 0.600 43.081 21.841 25 27.26313.798 0.334 −13.464 13.464 0.501 43.402 21.952 26 27.436 13.852 0.267−13.585 13.585 0.400 43.721 22.061 27 27.230 13.715 0.201 −13.515 13.5150.301 43.342 21.822 28 27.241 13.687 0.133 −13.554 13.554 0.200 43.36121.781 29 27.416 13.742 0.067 −13.674 13.674 0.101 43.683 21.892 3027.589 13.794 0.000 −13.794 13.794 0.000 44.002 22.001 31 27.600 13.766−0.067 −13.833 13.833 −0.100 44.022 21.961 32 27.664 13.766 −0.133−13.899 13.899 −0.200 44.142 21.971 33 27.837 13.818 −0.200 −14.01814.018 −0.300 44.462 22.081 34 27.955 13.844 −0.267 −14.111 14.111−0.400 44.682 22.141 35 28.074 13.870 −0.333 −14.203 14.203 −0.49944.903 22.202 36 28.030 13.815 −0.401 −14.215 14.215 −0.601 44.82222.111 37 28.042 13.788 −0.467 −14.254 14.254 −0.700 44.844 22.072 3828.106 13.786 −0.533 −14.320 14.320 −0.800 44.964 22.082 39 28.22413.812 −0.600 −14.412 14.412 −0.900 45.184 22.142 40 28.288 13.811−0.667 −14.477 14.477 −1.000 45.304 22.152 41 28.298 13.783 −0.733−14.516 14.516 −1.100 45.324 22.112 42 28.362 13.781 −0.800 −14.58114.581 −1.200 45.444 22.122 43 28.427 13.781 −0.866 −14.646 14.646−1.299 45.566 22.134 44 28.437 13.752 −0.933 −14.685 14.685 −1.40045.585 22.093 45 28.607 13.804 −0.999 −14.803 14.803 −1.499 45.90622.204 46 28.670 13.802 −1.067 −14.868 14.868 −1.600 46.026 22.213 4728.734 13.800 −1.133 −14.933 14.933 −1.700 46.146 22.223 48 28.79713.798 −1.200 −14.998 14.998 −1.800 46.266 22.233 49 28.808 13.771−1.266 −15.037 15.037 −1.899 46.287 22.194 50 28.923 13.795 −1.333−15.128 15.128 −2.000 46.506 22.253 51 28.829 13.715 −1.399 −15.11415.114 −2.099 46.327 22.114 52 28.996 13.765 −1.466 −15.231 15.231−2.200 46.646 22.223 53 29.007 13.737 −1.532 −15.270 15.270 −2.29946.667 22.184 54 29.069 13.735 −1.600 −15.335 15.335 −2.400 46.78622.193 55 29.028 13.681 −1.666 −15.347 15.347 −2.499 46.707 22.104 5629.142 13.705 −1.733 −15.438 15.438 −2.600 46.926 22.163 57 29.30913.755 −1.799 −15.554 15.554 −2.699 47.247 22.274 58 29.475 13.804−1.866 −15.670 15.670 −2.800 47.566 22.383 59 29.330 13.699 −1.932−15.631 15.631 −2.899 47.287 22.194 60 29.392 13.696 −1.999 −15.69615.696 −3.000 47.406 22.203 61 29.558 13.746 −2.065 −15.812 15.812−3.099 47.727 22.314 62 29.670 13.769 −2.133 −15.902 15.902 −3.20047.946 22.373 63 29.630 13.716 −2.199 −15.914 15.914 −3.299 47.86722.284 64 29.640 13.687 −2.266 −15.953 15.953 −3.400 47.886 22.243 6529.752 13.710 −2.333 −16.043 16.043 −3.500 48.106 22.303 66 29.91613.759 −2.399 −16.158 16.158 −3.600 48.426 22.413 67 29.825 13.680−2.465 −16.145 16.145 −3.699 48.247 22.274 68 29.988 13.728 −2.532−16.260 16.260 −3.800 48.566 22.383 69 30.151 13.776 −2.598 −16.37416.374 −3.899 48.887 22.494 70 30.160 13.747 −2.665 −16.413 16.413−4.000 48.906 22.453 71 30.170 13.719 −2.732 −16.451 16.451 −4.10048.926 22.413 72 30.332 13.767 −2.799 −16.565 16.565 −4.200 49.24622.523 73 30.394 13.765 −2.865 −16.629 16.629 −4.299 49.368 22.535 7430.302 13.685 −2.932 −16.617 16.617 −4.400 49.187 22.394 75 30.36313.683 −2.998 −16.681 16.681 −4.499 49.308 22.405 76 30.474 13.704−3.065 −16.769 16.769 −4.600 49.527 22.464 77 30.634 13.752 −3.131−16.883 16.883 −4.699 49.848 22.575 78 30.694 13.748 −3.198 −16.94616.946 −4.800 49.967 22.584 79 30.654 13.695 −3.264 −16.959 16.959−4.899 49.888 22.495 80 30.814 13.741 −3.331 −17.072 17.072 −5.00050.208 22.604 81 30.874 13.738 −3.397 −17.135 17.135 −5.099 50.32922.615 82 30.834 13.685 −3.464 −17.149 17.149 −5.200 50.248 22.524 8330.894 13.682 −3.530 −17.212 17.212 −5.299 50.369 22.535 84 31.05113.727 −3.597 −17.324 17.324 −5.400 50.688 22.644 85 31.160 13.749−3.663 −17.411 17.411 −5.499 50.909 22.705 86 31.169 13.720 −3.730−17.450 17.450 −5.600 50.928 22.664 87 31.180 13.692 −3.796 −17.48817.488 −5.699 50.949 22.625 88 31.336 13.736 −3.863 −17.599 17.599−5.800 51.268 22.734 89 31.443 13.757 −3.929 −17.686 17.686 −5.89951.488 22.795 90 31.549 13.777 −3.996 −17.772 17.772 −6.000 51.70622.853 91 31.704 13.821 −4.062 −17.883 17.883 −6.099 52.027 22.964 9231.713 13.792 −4.129 −17.921 17.921 −6.200 52.046 22.923 93 31.72313.764 −4.195 −17.959 17.959 −6.299 52.067 22.884 94 31.636 13.687−4.262 −17.949 17.949 −6.400 51.886 22.743 95 31.695 13.684 −4.327−18.011 18.011 −6.499 52.007 22.754 96 31.848 13.727 −4.395 −18.12118.121 −6.600 52.326 22.863 97 31.858 13.699 −4.460 −18.159 18.159−6.699 52.347 22.824 98 31.963 13.718 −4.527 −18.245 18.245 −6.80052.566 22.883 99 32.116 13.762 −4.593 −18.355 18.355 −6.899 52.88822.995 100 32.267 13.804 −4.660 −18.464 18.464 −7.000 53.207 23.104 Mean= 13.759 Mean = 22.234 Standard Deviation = 0.047 N = 100 N = 100

TABLE A-2 ANGLES OF SECOND INCIDENT LIGHT, SECOND REFLECTED LIGHT, ANDSECOND REFLECTIVE AXIS, FOR A FIRST EMBODIMENT SKEW MIRROR; WAVELENGTH =513 NM; AK174-200 RECORDING MEDIUM Angle Of Angle Of Angle of Angle ofIncidence Reflection Incidence of Reflection of Second Second Second ofSecond of Second Second Second Internal Reflective Internal IncidentReflected Incident Reflected Second Reflective Angle of Axis Angle AngleOf Light Light Light Light Axis Angle Reflection (INTERNAL, Incidence(INTERNAL, (INTERNAL, (EXTERNAL, (EXTERNAL, (EXTERNAL, (relativerelative (relative relative to relative to relative to relative torelative to surface to surface to surface reflective reflective surfacesurface to surface normal, in normal, normal, axis, in axis, in normal,in normal, in normal, in Measurement degrees) indegrees) in degrees)degrees) degrees) degrees) degrees) degrees) 1 25.273 13.603 1.933−11.670 11.670 2.900 39.821 21.361 2 25.341 13.604 1.866 −11.737 11.7372.800 39.942 21.371 3 25.466 13.633 1.800 −11.833 11.833 2.701 40.16321.432 4 25.645 13.689 1.733 −11.956 11.956 2.600 40.481 21.541 5 25.76913.718 1.667 −12.051 12.051 2.501 40.702 21.602 6 25.780 13.690 1.600−12.090 12.090 2.400 40.721 21.561 7 25.959 13.746 1.533 −12.213 12.2132.300 41.041 21.671 8 25.915 13.691 1.466 −12.224 12.224 2.200 40.96121.581 9 25.982 13.691 1.400 −12.291 12.291 2.100 41.081 21.591 1026.160 13.746 1.333 −12.413 12.413 2.000 41.400 21.700 11 26.171 13.7191.267 −12.452 12.452 1.900 41.420 21.660 12 26.181 13.691 1.200 −12.49112.491 1.800 41.439 21.620 13 26.249 13.691 1.134 −12.557 12.557 1.70141.560 21.631 14 26.259 13.663 1.067 −12.596 12.596 1.600 41.579 21.59015 26.438 13.719 1.001 −12.718 12.718 1.501 41.900 21.701 16 26.44813.691 0.933 −12.757 12.757 1.400 41.919 21.660 17 26.515 13.691 0.867−12.824 12.824 1.301 42.040 21.671 18 26.636 13.718 0.800 −12.918 12.9181.200 42.259 21.730 19 26.592 13.663 0.734 −12.929 12.929 1.101 42.18021.641 20 26.769 13.718 0.667 −13.051 13.051 1.000 42.500 21.750 2126.780 13.690 0.601 −13.090 13.090 0.901 42.520 21.711 22 26.845 13.6890.533 −13.156 13.156 0.800 42.639 21.720 23 26.912 13.690 0.467 −13.22213.222 0.701 42.760 21.731 24 26.977 13.689 0.400 −13.289 13.289 0.60042.879 21.740 25 26.989 13.661 0.334 −13.327 13.327 0.501 42.900 21.70126 27.108 13.687 0.266 −13.421 13.421 0.399 43.118 21.759 27 27.22913.715 0.201 −13.514 13.514 0.301 43.340 21.821 28 27.240 13.686 0.133−13.553 13.553 0.200 43.359 21.780 29 27.360 13.714 0.067 −13.646 13.6460.101 43.580 21.841 30 27.425 13.713 0.000 −13.713 13.713 0.000 43.70021.850 31 27.490 13.712 −0.066 −13.778 13.778 −0.099 43.820 21.861 3227.555 13.711 −0.133 −13.844 13.844 −0.200 43.939 21.870 33 27.56513.683 −0.200 −13.883 13.883 −0.300 43.959 21.830 34 27.630 13.682−0.267 −13.949 13.949 −0.400 44.079 21.840 35 27.750 13.709 −0.333−14.041 14.041 −0.499 44.300 21.901 36 27.760 13.680 −0.400 −14.08014.080 −0.600 44.319 21.860 37 27.825 13.680 −0.466 −14.146 14.146−0.699 44.440 21.871 38 27.889 13.678 −0.533 −14.211 14.211 −0.80044.559 21.880 39 28.007 13.703 −0.600 −14.303 14.303 −0.900 44.77821.939 40 28.017 13.675 −0.667 −14.342 14.342 −1.000 44.798 21.899 4128.135 13.701 −0.733 −14.434 14.434 −1.100 45.018 21.959 42 28.25313.726 −0.800 −14.526 14.526 −1.200 45.238 22.019 43 28.264 13.699−0.866 −14.565 14.565 −1.299 45.259 21.980 44 28.274 13.670 −0.933−14.604 14.604 −1.400 45.278 21.939 45 28.338 13.669 −0.999 −14.66914.669 −1.499 45.399 21.950 46 28.455 13.694 −1.067 −14.761 14.761−1.600 45.619 22.010 47 28.572 13.719 −1.133 −14.852 14.852 −1.70045.839 22.070 48 28.635 13.718 −1.200 −14.917 14.917 −1.800 45.95922.080 49 28.646 13.690 −1.267 −14.956 14.956 −1.900 45.979 22.040 5028.709 13.688 −1.333 −15.021 15.021 −2.000 46.099 22.050 51 28.72013.660 −1.399 −15.060 15.060 −2.099 46.120 22.011 52 28.835 13.684−1.466 −15.151 15.151 −2.200 46.339 22.070 53 28.899 13.683 −1.532−15.216 15.216 −2.299 46.460 22.081 54 29.013 13.707 −1.600 −15.30715.307 −2.400 46.679 22.140 55 29.024 13.679 −1.666 −15.345 15.345−2.499 46.700 22.101 56 29.087 13.677 −1.733 −15.410 15.410 −2.60046.819 22.110 57 29.150 13.675 −1.799 −15.474 15.474 −2.699 46.94022.121 58 29.264 13.699 −1.866 −15.565 15.565 −2.800 47.159 22.180 5929.326 13.697 −1.932 −15.629 15.629 −2.899 47.280 22.191 60 29.38813.694 −1.999 −15.694 15.694 −3.000 47.399 22.200 61 29.502 13.718−2.065 −15.784 15.784 −3.099 47.620 22.261 62 29.667 13.767 −2.133−15.900 15.900 −3.200 47.939 22.370 63 29.678 13.739 −2.199 −15.93815.938 −3.299 47.960 22.331 64 29.790 13.762 −2.266 −16.028 16.028−3.400 48.180 22.390 65 29.647 13.657 −2.333 −15.990 15.990 −3.50047.900 22.200 66 29.760 13.680 −2.399 −16.079 16.079 −3.600 48.12022.260 67 29.822 13.678 −2.465 −16.143 16.143 −3.699 48.241 22.271 6829.882 13.675 −2.532 −16.207 16.207 −3.800 48.360 22.280 69 29.94413.672 −2.599 −16.271 16.271 −3.900 48.480 22.290 70 30.056 13.695−2.665 −16.361 16.361 −4.000 48.700 22.350 71 30.066 13.667 −2.732−16.399 16.399 −4.100 48.721 22.311 72 30.229 13.715 −2.799 −16.51416.514 −4.200 49.041 22.421 73 30.290 13.713 −2.865 −16.577 16.577−4.299 49.162 22.432 74 30.349 13.709 −2.932 −16.641 16.641 −4.40049.280 22.440 75 30.360 13.681 −2.998 −16.679 16.679 −4.499 49.30122.401 76 30.420 13.677 −3.065 −16.742 16.742 −4.600 49.420 22.410 7730.531 13.700 −3.131 −16.831 16.831 −4.699 49.641 22.471 78 30.59013.696 −3.198 −16.894 16.894 −4.800 49.760 22.480 79 30.651 13.694−3.264 −16.957 16.957 −4.899 49.881 22.491 80 30.710 13.690 −3.331−17.021 17.021 −5.000 50.000 22.500 81 30.820 13.712 −3.397 −17.10917.109 −5.099 50.221 22.561 82 30.830 13.683 −3.464 −17.147 17.147−5.200 50.240 22.520 83 30.939 13.705 −3.530 −17.235 17.235 −5.29950.461 22.581 84 30.949 13.676 −3.597 −17.273 17.273 −5.400 50.48022.540 85 31.009 13.673 −3.663 −17.336 17.336 −5.499 50.602 22.552 8631.068 13.669 −3.730 −17.399 17.399 −5.600 50.721 22.561 87 31.22513.714 −3.797 −17.511 17.511 −5.700 51.041 22.671 88 31.284 13.710−3.863 −17.573 17.573 −5.800 51.161 22.681 89 31.293 13.682 −3.929−17.611 17.611 −5.900 51.181 22.641 90 31.352 13.678 −3.996 −17.67417.674 −6.000 51.302 22.651 91 31.460 13.699 −4.062 −17.761 17.761−6.099 51.522 22.712 92 31.517 13.694 −4.129 −17.823 17.823 −6.20051.641 22.721 93 31.528 13.667 −4.195 −17.861 17.861 −6.299 51.66222.682 94 31.682 13.710 −4.262 −17.972 17.972 −6.400 51.981 22.791 9531.692 13.682 −4.327 −18.010 18.010 −6.499 52.002 22.752 96 31.79813.701 −4.395 −18.096 18.096 −6.600 52.221 22.811 97 31.904 13.722−4.460 −18.182 18.182 −6.699 52.442 22.872 98 31.913 13.693 −4.527−18.220 18.220 −6.800 52.461 22.831 99 31.970 13.689 −4.593 −18.28218.282 −6.899 52.582 22.842 100 32.075 13.707 −4.660 −18.368 18.368−7.000 52.801 22.901 Mean = 13.693 Mean = 22.110 Standard Deviation =0.025 N = 100 N = 100

TABLE A-3 ANGLES OF FIRST INCIDENT LIGHT, FIRST REFLECTED LIGHT, ANDFIRST REFLECTIVE AXIS, FOR A SECOND EMBODIMENT SKEW MIRROR; WAVELENGTH =532 NM; AK233-200 RECORDING MEDIUM First Internal Angle Of Angle OfAngle of Angle of First Internal Angle Of Incidence Reflection Incidenceof Reflection Angle of Incidence of First of First First of First FirstReflection First of First Incident Reflected Incident ReflectedReflective of First Reflective Incident Light Light Light Light AxisAngle Reflected Axis Angle Light (internal, (internal, (EXTERNAL,(EXTERNAL, (EXTERNAL, Light (relative (relative relative to relative torelative to relative relative to (relative to to surface to surfacereflective reflective surface to surface surface surface normal, normal,normal, axis, in axis, normal, in normal, normal, in Measurement indegrees) indegrees) in degrees) degrees) in degrees) degrees) indegrees) degrees) 1 31.836 14.585 −2.665 −17.250 17.250 −4.000 52.30024.150 2 31.941 14.604 −2.732 −17.336 17.336 −4.100 52.520 24.210 331.998 14.600 −2.799 −17.398 17.398 −4.200 52.640 24.220 4 32.103 14.619−2.865 −17.484 17.484 −4.299 52.861 24.281 5 32.160 14.614 −2.932−17.546 17.546 −4.400 52.980 24.290 6 32.217 14.610 −2.998 −17.60717.607 −4.499 53.101 24.301 7 32.321 14.628 −3.065 −17.693 17.693 −4.60053.320 24.360 8 32.378 14.623 −3.131 −17.754 17.754 −4.699 53.441 24.3719 32.433 14.618 −3.198 −17.816 17.816 −4.800 53.560 24.380 10 32.49014.613 −3.264 −17.877 17.877 −4.899 53.681 24.391 11 32.546 14.607−3.331 −17.938 17.938 −5.000 53.800 24.400 12 32.602 14.603 −3.397−18.000 18.000 −5.099 53.921 24.411 13 32.704 14.620 −3.464 −18.08418.084 −5.200 54.140 24.470 14 32.760 14.615 −3.530 −18.145 18.145−5.299 54.261 24.481 15 32.815 14.609 −3.597 −18.206 18.206 −5.40054.380 24.490 16 32.871 14.604 −3.664 −18.267 18.267 −5.500 54.50024.500 17 32.926 14.598 −3.730 −18.328 18.328 −5.600 54.620 24.510 1833.027 14.616 −3.796 −18.412 18.412 −5.699 54.841 24.571 19 33.08214.609 −3.863 −18.472 18.472 −5.800 54.960 24.580 20 33.137 14.604−3.929 −18.533 18.533 −5.899 55.081 24.591 21 33.191 14.598 −3.996−18.594 18.594 −6.000 55.200 24.600 22 33.291 14.615 −4.062 −18.67718.677 −6.099 55.421 24.661 23 33.345 14.608 −4.129 −18.737 18.737−6.200 55.540 24.670 24 33.400 14.603 −4.195 −18.797 18.797 −6.29955.661 24.681 25 33.498 14.618 −4.262 −18.880 18.880 −6.400 55.88024.740 26 33.552 14.612 −4.327 −18.940 18.940 −6.499 56.001 24.751 2733.605 14.605 −4.395 −19.000 19.000 −6.600 56.120 24.760 28 33.65914.600 −4.460 −19.060 19.060 −6.699 56.241 24.771 29 33.757 14.615−4.527 −19.142 19.142 −6.800 56.460 24.830 30 33.810 14.608 −4.593−19.201 19.201 −6.899 56.580 24.841 31 33.862 14.601 −4.660 −19.26119.261 −7.000 56.699 24.850 32 33.916 14.595 −4.726 −19.321 19.321−7.099 56.820 24.861 33 34.012 14.609 −4.793 −19.402 19.402 −7.20057.039 24.920 34 34.064 14.603 −4.859 −19.462 19.462 −7.299 57.16024.931 35 34.116 14.595 −4.926 −19.521 19.521 −7.400 57.279 24.940 3634.169 14.588 −4.992 −19.580 19.580 −7.500 57.399 24.950 37 34.26414.603 −5.058 −19.661 19.661 −7.600 57.619 25.010 38 34.316 14.596−5.124 −19.720 19.720 −7.699 57.740 25.021 39 34.368 14.588 −5.191−19.779 19.779 −7.800 57.860 25.030 40 34.462 14.602 −5.257 −19.86019.860 −7.900 58.080 25.090 41 34.513 14.595 −5.324 −19.918 19.918−8.000 58.199 25.100 42 34.606 14.608 −5.390 −19.998 19.998 −8.10058.419 25.160 43 34.699 14.622 −5.456 −20.078 20.078 −8.200 58.63925.220 44 34.750 14.614 −5.522 −20.136 20.136 −8.299 58.760 25.231 4534.842 14.626 −5.589 −20.216 20.216 −8.401 58.978 25.289 46 34.89314.619 −5.655 −20.274 20.274 −8.500 59.100 25.300 47 34.943 14.611−5.721 −20.332 20.332 −8.600 59.220 25.310 48 35.035 14.624 −5.787−20.411 20.411 −8.699 59.441 25.371 49 35.084 14.615 −5.854 −20.46920.469 −8.800 59.560 25.380 50 35.134 14.607 −5.919 −20.527 20.527−8.899 59.681 25.391 51 35.224 14.619 −5.986 −20.605 20.605 −9.00059.900 25.450 52 35.273 14.611 −6.052 −20.662 20.662 −9.099 60.02125.461 53 35.321 14.601 −6.119 −20.720 20.720 −9.200 60.140 25.470 5435.411 14.613 −6.184 −20.798 20.798 −9.299 60.361 25.531 55 35.45914.604 −6.251 −20.855 20.855 −9.400 60.479 25.540 56 35.548 14.616−6.316 −20.932 20.932 −9.499 60.700 25.601 57 35.595 14.606 −6.383−20.989 20.989 −9.600 60.819 25.610 58 35.683 14.617 −6.449 −21.06621.066 −9.699 61.040 25.671 59 35.731 14.608 −6.516 −21.123 21.123−9.800 61.159 25.680 60 35.817 14.618 −6.582 −21.200 21.200 −9.90061.379 25.740 61 35.865 14.608 −6.648 −21.256 21.256 −10.000 61.49925.750 62 35.951 14.618 −6.714 −21.332 21.332 −10.100 61.719 25.810 6335.997 14.609 −6.780 −21.389 21.389 −10.200 61.839 25.820 64 36.08314.619 −6.845 −21.464 21.464 −10.299 62.060 25.881 65 36.168 14.628−6.912 −21.540 21.540 −10.400 62.279 25.940 66 36.214 14.618 −6.977−21.596 21.596 −10.499 62.400 25.951 67 36.298 14.627 −7.044 −21.67121.671 −10.600 62.619 26.010 68 36.343 14.617 −7.110 −21.726 21.726−10.699 62.739 26.020 69 36.426 14.625 −7.176 −21.801 21.801 −10.80062.958 26.079 70 36.471 14.615 −7.242 −21.856 21.856 −10.899 63.07926.090 71 36.553 14.623 −7.308 −21.931 21.931 −11.000 63.298 26.149 7236.635 14.631 −7.373 −22.004 22.004 −11.099 63.519 26.210 73 36.67914.620 −7.440 −22.060 22.060 −11.200 63.638 26.219 74 36.761 14.628−7.505 −22.133 22.133 −11.299 63.859 26.280 75 36.804 14.616 −7.572−22.188 22.188 −11.400 63.978 26.289 76 36.885 14.624 −7.637 −22.26122.261 −11.499 64.199 26.350 77 36.964 14.630 −7.704 −22.334 22.334−11.600 64.418 26.409 78 37.007 14.619 −7.769 −22.388 22.388 −11.69964.539 26.420 79 37.086 14.625 −7.836 −22.461 22.461 −11.800 64.75826.479 80 37.164 14.631 −7.901 −22.533 22.533 −11.900 64.978 26.539 8137.241 14.637 −7.967 −22.604 22.604 −12.000 65.198 26.599 82 37.28414.625 −8.033 −22.658 22.658 −12.100 65.318 26.609 83 37.360 14.630−8.099 −22.729 22.729 −12.200 65.537 26.669 84 37.436 14.636 −8.165−22.800 22.800 −12.300 65.757 26.729 85 37.512 14.640 −8.231 −22.87122.871 −12.400 65.977 26.789 86 37.553 14.629 −8.296 −22.924 22.924−12.499 66.098 26.800 87 37.627 14.633 −8.362 −22.995 22.995 −12.60066.317 26.859 88 37.702 14.637 −8.427 −23.064 23.064 −12.699 66.53826.920 89 37.774 14.640 −8.494 −23.134 23.134 −12.800 66.756 26.978 9037.848 14.645 −8.559 −23.203 23.203 −12.899 66.978 27.040 91 37.92014.648 −8.625 −23.273 23.273 −13.000 67.197 27.099 92 37.960 14.635−8.690 −23.325 23.325 −13.099 67.318 27.110 93 38.031 14.637 −8.756−23.394 23.394 −13.200 67.537 27.169 94 38.102 14.640 −8.822 −23.46223.462 −13.300 67.757 27.229 95 38.172 14.642 −8.888 −23.530 23.530−13.400 67.977 27.289 96 38.242 14.644 −8.953 −23.597 23.597 −13.49968.197 27.349 97 38.310 14.645 −9.019 −23.664 23.664 −13.600 68.41527.408 98 38.379 14.647 −9.084 −23.731 23.731 −13.699 68.636 27.469 9938.446 14.648 −9.150 −23.798 23.798 −13.800 68.855 27.528 100 38.51414.649 −9.215 −23.864 23.864 −13.899 69.076 27.589 101 38.610 14.664−9.281 −23.946 23.946 −14.000 69.395 27.698 Mean = 14.618 Mean = 25.594Standard Deviation = 0.016 N = 101 N = 101

TABLE A-4 ANGLES OF SECOND INCIDENT LIGHT, SECOND REFLECTED LIGHT, ANDSECOND REFLECTIVE AXIS, FOR A SECOND EMBODIMENT SKEW MIRROR; WAVELENGTH= 513 NM; AK233-200 RECORDING MEDIUM Angle of Angle Of Angle Of Angle OfAngle of Angle of Reflection Incidence Incidence Reflection IncidenceReflection Second of Second Second of Second of Second of Second ofSecond of Second Reflective Reflected Reflective Incident IncidentReflected Incident Reflected Axis Light Axis Light Light Light LightLight Angle (internal, Angle (internal, (internal, (internal, (EXTERNAL,(EXTERNAL, (EXTERNAL, relative (relative relative relative relative torelative relative to relative to surface to surface to surface toreflective reflective to surface surface to surface normal, in normal,normal, axis, axis, normal, in normal, in normal, in Measurementdegrees) indegrees) in degrees) degrees) in degrees) degrees) degrees)degrees) 1 31.836 14.585 −2.665 −17.250 17.250 −4.000 52.300 24.150 231.941 14.604 −2.732 −17.336 17.336 −4.100 52.520 24.210 3 32.022 14.612−2.799 −17.410 17.410 −4.200 52.690 24.245 4 32.080 14.608 −2.865−17.472 17.472 −4.299 52.811 24.256 5 32.160 14.614 −2.932 −17.54617.546 −4.400 52.980 24.290 6 32.240 14.621 −2.998 −17.619 17.619 −4.49953.150 24.326 7 32.297 14.616 −3.065 −17.681 17.681 −4.600 53.270 24.3358 32.378 14.623 −3.131 −17.754 17.754 −4.699 53.441 24.371 9 32.43414.618 −3.198 −17.816 17.816 −4.800 53.561 24.381 10 32.514 14.625−3.264 −17.889 17.889 −4.899 53.732 24.417 11 32.570 14.619 −3.331−17.950 17.950 −5.000 53.851 24.426 12 32.626 14.615 −3.397 −18.01118.011 −5.099 53.972 24.437 13 32.705 14.620 −3.464 −18.084 18.084−5.200 54.141 24.471 14 32.737 14.604 −3.530 −18.134 18.134 −5.29954.212 24.457 15 32.816 14.610 −3.597 −18.207 18.207 −5.400 54.38224.491 16 32.872 14.605 −3.663 −18.267 18.267 −5.500 54.503 24.502 1732.950 14.610 −3.730 −18.340 18.340 −5.600 54.672 24.536 18 33.00614.605 −3.796 −18.401 18.401 −5.699 54.794 24.548 19 33.060 14.598−3.863 −18.461 18.461 −5.800 54.912 24.556 20 33.137 14.604 −3.929−18.533 18.533 −5.899 55.082 24.592 21 33.215 14.609 −3.996 −18.60518.605 −6.000 55.252 24.626 22 33.292 14.615 −4.062 −18.677 18.677−6.099 55.423 24.662 23 33.346 14.608 −4.129 −18.737 18.737 −6.20055.541 24.671 24 33.423 14.614 −4.195 −18.809 18.809 −6.299 55.71324.707 25 33.477 14.608 −4.262 −18.869 18.869 −6.400 55.833 24.717 2633.554 14.613 −4.327 −18.941 18.941 −6.499 56.004 24.753 27 33.60714.606 −4.395 −19.001 19.001 −6.600 56.123 24.762 28 33.683 14.611−4.460 −19.072 19.072 −6.699 56.294 24.798 29 33.758 14.615 −4.527−19.143 19.143 −6.800 56.463 24.832 30 33.812 14.609 −4.593 −19.20219.202 −6.899 56.584 24.843 31 33.886 14.613 −4.660 −19.273 19.273−7.000 56.752 24.876 32 33.939 14.607 −4.726 −19.333 19.333 −7.09956.874 24.888 33 33.992 14.599 −4.793 −19.392 19.392 −7.200 56.99424.897 34 34.067 14.604 −4.859 −19.463 19.463 −7.299 57.165 24.933 3534.141 14.608 −4.926 −19.533 19.533 −7.400 57.335 24.968 36 34.19214.600 −4.992 −19.592 19.592 −7.500 57.454 24.977 37 34.266 14.604−5.058 −19.662 19.662 −7.600 57.624 25.012 38 34.318 14.597 −5.124−19.721 19.721 −7.699 57.745 25.023 39 34.391 14.600 −5.191 −19.79119.791 −7.800 57.915 25.058 40 34.443 14.593 −5.257 −19.850 19.850−7.900 58.036 25.068 41 34.258 14.467 −5.324 −19.791 19.791 −8.00057.606 24.803 42 34.418 14.514 −5.390 −19.904 19.904 −8.100 57.97724.939 43 34.576 14.560 −5.456 −20.016 20.016 −8.200 58.348 25.074 4434.733 14.606 −5.522 −20.127 20.127 −8.299 58.719 25.210 45 34.84614.629 −5.589 −20.217 20.217 −8.401 58.988 25.294 46 34.897 14.621−5.654 −20.276 20.276 −8.500 59.109 25.305 47 34.967 14.623 −5.721−20.344 20.344 −8.600 59.279 25.340 48 35.018 14.615 −5.787 −20.40220.402 −8.699 59.400 25.351 49 35.108 14.627 −5.854 −20.481 20.481−8.800 59.618 25.409 50 35.137 14.609 −5.919 −20.528 20.528 −8.89959.690 25.396 51 35.207 14.610 −5.986 −20.596 20.596 −9.000 59.85925.430 52 35.277 14.612 −6.052 −20.664 20.664 −9.099 60.030 25.466 5335.345 14.613 −6.119 −20.732 20.732 −9.200 60.198 25.499 54 35.41414.615 −6.184 −20.799 20.799 −9.299 60.368 25.535 55 35.482 14.615−6.251 −20.866 20.866 −9.400 60.536 25.568 56 35.551 14.617 −6.316−20.934 20.934 −9.499 60.708 25.605 57 35.618 14.617 −6.383 −21.00121.001 −9.600 60.876 25.638 58 35.666 14.608 −6.449 −21.058 21.058−9.699 60.996 25.649 59 35.753 14.619 −6.516 −21.134 21.134 −9.80061.216 25.708 60 35.820 14.619 −6.582 −21.201 21.201 −9.900 61.38525.743 61 35.887 14.619 −6.648 −21.267 21.267 −10.000 61.555 25.778 6235.954 14.620 −6.713 −21.334 21.334 −10.100 61.727 25.814 63 36.02014.620 −6.780 −21.400 21.400 −10.200 61.897 25.849 64 36.067 14.611−6.845 −21.456 21.456 −10.299 62.017 25.859 65 36.170 14.629 −6.912−21.541 21.541 −10.400 62.286 25.943 66 36.217 14.620 −6.977 −21.59721.597 −10.499 62.407 25.954 67 36.282 14.619 −7.044 −21.663 21.663−10.600 62.577 25.989 68 36.365 14.628 −7.110 −21.737 21.737 −10.69962.798 26.050 69 36.429 14.627 −7.176 −21.803 21.803 −10.800 62.96726.084 70 36.475 14.617 −7.242 −21.858 21.858 −10.899 63.089 26.095 7136.557 14.625 −7.308 −21.933 21.933 −11.000 63.309 26.155 72 36.62114.624 −7.373 −21.997 21.997 −11.099 63.480 26.191 73 36.665 14.612−7.440 −22.053 22.053 −11.200 63.599 26.200 74 36.746 14.620 −7.505−22.126 22.126 −11.299 63.819 26.260 75 36.826 14.627 −7.572 −22.19922.199 −11.400 64.037 26.319 76 36.888 14.626 −7.637 −22.263 22.263−11.499 64.209 26.355 77 36.950 14.623 −7.704 −22.327 22.327 −11.60064.379 26.390 78 37.029 14.630 −7.769 −22.399 22.399 −11.699 64.60026.451 79 37.107 14.636 −7.836 −22.472 22.472 −11.800 64.819 26.510 8037.185 14.642 −7.901 −22.543 22.543 −11.900 65.039 26.570 81 37.22814.630 −7.967 −22.598 22.598 −12.000 65.159 26.580 82 37.305 14.636−8.033 −22.669 22.669 −12.100 65.380 26.640 83 37.364 14.633 −8.099−22.731 22.731 −12.200 65.549 26.675 84 37.440 14.638 −8.165 −22.80322.803 −12.300 65.770 26.735 85 37.499 14.634 −8.231 −22.865 22.865−12.400 65.940 26.770 86 37.557 14.631 −8.296 −22.926 22.926 −12.49966.111 26.806 87 37.632 14.635 −8.362 −22.997 22.997 −12.600 66.33026.865 88 37.706 14.639 −8.427 −23.067 23.067 −12.699 66.551 26.926 8937.779 14.643 −8.494 −23.136 23.136 −12.800 66.770 26.985 90 37.85214.647 −8.559 −23.206 23.206 −12.899 66.991 27.046 91 37.908 14.641−8.625 −23.266 23.266 −13.000 67.159 27.080 92 37.980 14.645 −8.690−23.335 23.335 −13.099 67.380 27.141 93 38.051 14.647 −8.756 −23.40423.404 −13.200 67.599 27.200 94 38.121 14.650 −8.822 −23.472 23.472−13.300 67.819 27.260 95 38.176 14.644 −8.888 −23.532 23.532 −13.40067.989 27.295 96 38.245 14.646 −8.953 −23.599 23.599 −13.499 68.20827.355 97 38.314 14.647 −9.019 −23.666 23.666 −13.600 68.427 27.414 9838.398 14.657 −9.084 −23.741 23.741 −13.699 68.697 27.499 99 38.46514.657 −9.150 −23.808 23.808 −13.800 68.916 27.558 100 38.517 14.651−9.215 −23.866 23.866 −13.899 69.087 27.594 101 38.598 14.658 −9.281−23.940 23.940 −14.000 69.355 27.678 Mean = 14.617 Mean = 25.593Standard Deviation = 0.025 N = 101 N = 101

TABLE A-5 RECORDING BEAM ANGLES FOR A METHOD OF MAKING THE FIRSTEMBODIMENT SKEW MIRROR; SKEW AXIS ANGLE = 13.726° RELATIVE TO SURFACENORMAL Internal Angle of First Recording Beam Second Recording BeamInternal Angle of First Second Recording Magnitude of Angle Angle AngleRecording Beam Beam Relative To Difference From HOLGRAM (internal,relative to surface normal) Relative To Skew Axis Skew Axis PreviousHologram 1 53.218 154.234 39.492 140.508 2 53.309 154.143 39.583 140.4170.091 3 53.400 154.052 39.674 140.326 0.091 4 53.491 153.961 39.765140.235 0.091 5 53.581 153.871 39.855 140.145 0.091 6 53.672 153.78039.946 140.054 0.090 7 53.762 153.690 40.036 139.964 0.090 8 53.852153.600 40.126 139.874 0.090 9 53.942 153.510 40.216 139.784 0.090 1054.031 153.421 40.305 139.695 0.090 11 54.121 153.331 40.395 139.6050.090 12 54.210 153.242 40.484 139.516 0.089 13 54.300 153.152 40.574139.426 0.089 14 54.389 153.063 40.663 139.337 0.089 15 54.478 152.97440.752 139.248 0.089 16 54.567 152.885 40.841 139.159 0.089 17 54.655152.797 40.929 139.071 0.089 18 54.744 152.708 41.018 138.982 0.089 1954.832 152.620 41.106 138.894 0.088 20 54.920 152.532 41.194 138.8060.088 21 55.008 152.444 41.282 138.718 0.088 22 55.096 152.356 41.370138.630 0.088 23 55.184 152.268 41.458 138.542 0.088 24 55.271 152.18141.545 138.455 0.088 25 55.359 152.093 41.633 138.367 0.087 26 55.446152.006 41.720 138.280 0.087 27 55.533 151.919 41.807 138.193 0.087 2855.620 151.832 41.894 138.106 0.087 29 55.707 151.745 41.981 138.0190.087 30 55.794 151.658 42.068 137.932 0.087 31 55.881 151.571 42.155137.845 0.087 32 55.967 151.485 42.241 137.759 0.086 33 56.053 151.39942.327 137.673 0.086 34 56.139 151.313 42.413 137.587 0.086 35 56.225151.227 42.499 137.501 0.086 36 56.311 151.141 42.585 137.415 0.086 3756.397 151.055 42.671 137.329 0.086 38 56.483 150.969 42.757 137.2430.086 39 56.568 150.884 42.842 137.158 0.086 40 56.654 150.798 42.928137.072 0.085 41 56.739 150.713 43.013 136.987 0.085 42 56.824 150.62843.098 136.902 0.085 43 56.909 150.543 43.183 136.817 0.085 44 56.994150.458 43.268 136.732 0.085 45 57.079 150.373 43.353 136.647 0.085 4657.163 150.289 43.437 136.563 0.085 47 57.248 150.204 43.522 136.4780.085 48 57.332 150.120 43.606 136.394 0.084

TABLE A-6 RECORDING BEAM ANGLES FOR A METHOD OF MAKING THE SECONDEMBODIMENT SKEW MIRROR; SKEW AXIS ANGLE = 14.618° RELATIVE TO SURFACENORMAL Internal Angle of First Recording Beam Second Recording BeamInternal Angle of First Second Recording Magnitude of Angle Angle AngleRecording Beam Beam Relative To Difference From HOLGRAM (internal,relative to surface normal) Relative To Skew Axis Skew Axis PreviousHologram 1 55.913 153.323 41.295 138.705 2 56.008 153.228 41.390 138.6100.095 3 56.102 153.134 41.484 138.516 0.094 4 56.196 153.040 41.578138.422 0.094 5 56.290 152.946 41.672 138.328 0.094 6 56.384 152.85241.766 138.234 0.094 7 56.477 152.759 41.859 138.141 0.093 8 56.571152.665 41.953 138.047 0.094 9 56.664 152.572 42.046 137.954 0.093 1056.757 152.479 42.139 137.861 0.093 11 56.849 152.387 42.231 137.7690.092 12 56.942 152.294 42.324 137.676 0.093 13 57.034 152.202 42.416137.584 0.092 14 57.127 152.109 42.509 137.491 0.093 15 57.219 152.01742.601 137.399 0.092 16 57.311 151.925 42.693 137.307 0.092 17 57.402151.834 42.784 137.216 0.091 18 57.494 151.742 42.876 137.124 0.092 1957.585 151.651 42.967 137.033 0.091 20 57.676 151.560 43.058 136.9420.091 21 57.767 151.469 43.149 136.851 0.091 22 57.858 151.378 43.240136.760 0.091 23 57.949 151.287 43.331 136.669 0.091 24 58.040 151.19643.422 136.578 0.091 25 58.130 151.106 43.512 136.488 0.090 26 58.220151.016 43.602 136.398 0.090 27 58.310 150.926 43.692 136.308 0.090 2858.400 150.836 43.782 136.218 0.090 29 58.490 150.746 43.872 136.1280.090 30 58.579 150.657 43.961 136.039 0.089 31 58.669 150.567 44.051135.949 0.090 32 58.758 150.478 44.140 135.860 0.089 33 58.847 150.38944.229 135.771 0.089 34 58.936 150.300 44.318 135.682 0.089 35 59.025150.211 44.407 135.593 0.089 36 59.113 150.123 44.495 135.505 0.088 3759.202 150.034 44.584 135.416 0.089 38 59.290 149.946 44.672 135.3280.088 39 59.378 149.858 44.760 135.240 0.088 40 59.466 149.770 44.848135.152 0.088 41 59.554 149.682 44.936 135.064 0.088 42 59.642 149.59445.024 134.976 0.088 43 59.730 149.506 45.112 134.888 0.088 44 59.817149.419 45.199 134.801 0.087 45 59.904 149.332 45.286 134.714 0.087 4659.991 149.245 45.373 134.627 0.087 47 60.078 149.158 45.460 134.5400.087 48 60.165 149.071 45.547 134.453 0.087 49 60.252 148.984 45.634134.366 0.087

We claim:
 1. Apparatus comprising: a grating structure residing in agrating medium, wherein: the grating structure is configured to reflectfirst incident light, the first incident light being incident upon thegrating medium at a specific site and having a first wavelength and afirst internal angle of incidence relative to a surface normal of thegrating medium; the first incident light is principally reflected by thegrating medium as first reflected light, the first reflected lighthaving the first wavelength and a first internal angle of reflectionrelative to the surface normal; the first incident light and the firstreflected light are bisected by a first reflective axis having a firstreflective axis angle relative to the surface normal; the gratingstructure is further configured to reflect second incident light, thesecond incident light being incident on the grating medium at thespecific site and having a second wavelength and a second internal angleof incidence relative to the surface normal; the second incident lightis principally reflected by the grating medium as second reflectedlight, the second reflected light having the second wavelength and asecond internal angle of reflection relative to the surface normal; thesecond incident light and the second reflected light are bisected by asecond reflective axis having a second reflective axis angle relative tothe surface normal; the first and second reflective axis angles are eachnon-zero relative to the surface normal; the first wavelength differsfrom the second wavelength; and the first reflective axis angle differsfrom the second reflective axis angle.
 2. The apparatus of claim 1,wherein the first reflective axis angle differs from the secondreflective axis angle by 0.25 degrees or less.
 3. The apparatus of claim2, wherein the first reflective axis angle relative to the surfacenormal is at least 1.0 degree.
 4. The apparatus of claim 1, wherein thefirst reflective axis angle relative to the surface normal is at least1.0 degree.
 5. The apparatus of claim 1, wherein the first incidentlight is offset from the first reflective axis by at least 1.0 degree.6. The apparatus of claim 1, wherein: the grating structure comprises aplurality of volume holograms; each of the volume holograms in theplurality of volume holograms spatially overlaps at least one othervolume hologram in the plurality of volume holograms; and the gratingmedium is at least 70 μm thick.
 7. The apparatus of claim 6, wherein:the plurality of volume holograms includes at least four volumeholograms.
 8. The apparatus of claim 1, wherein: the grating structurecomprises at least 9 volume holograms; each of the at least 9 volumeholograms at least partially spatially overlaps all others of the atleast 9 volume holograms; and the grating medium is at least 200 μmthick.
 9. The apparatus defined in claim 1, wherein the first wavelengthdiffers from the second wavelength by a wave fraction of at least 0.005.10. The apparatus defined in claim 9, wherein the first reflective axisangle relative to the surface normal is at least 1.0 degree.
 11. Theapparatus defined in claim 9, wherein the first internal angle ofincidence is the same as the second internal angle of incidence.
 12. Theapparatus defined in claim 9, wherein each of the first internal angleof incidence and the second internal angle of incidence includesmultiple angles spanning a range.
 13. The apparatus defined in claim 1,wherein the first internal angle of incidence is the same as the secondinternal angle of incidence.
 14. The apparatus defined in claim 13,wherein each of the first internal angle of incidence and the secondinternal angle of incidence includes multiple angles spanning a range.15. Apparatus comprising: a grating structure residing in a gratingmedium, wherein: the grating structure is configured to reflect firstincident light, the first incident light being incident upon the gratingmedium at a specific site and having a first internal angle of incidencerelative to a surface normal of the grating medium; the first incidentlight is principally reflected by the grating medium as first reflectedlight, the first reflected light having a first internal angle ofreflection relative to the surface normal; the first incident light andthe first reflected light are bisected by a first reflective axis havinga first reflective axis angle relative to the surface normal; thegrating structure is further configured to reflect second incidentlight, the second incident light being incident on the grating medium atthe specific site and having a second internal angle of incidencerelative to the surface normal; the second incident light is principallyreflected by the grating medium as second reflected light, the secondreflected light having a second internal angle of reflection relative tothe surface normal; the second incident light and the second reflectedlight are bisected by a second reflective axis having a secondreflective axis angle relative to the surface normal; the first incidentlight has the same wavelength as the first reflected light; the secondincident light has the same wavelength as the second reflected light;the first internal angle of incidence differs from the second internalangle of incidence by a multiple of Δθ_(B), wherein Δθ_(B) is an angularBragg selectivity calculated for an incident light angle of incidenceresiding at a midpoint between the first incident light and the secondincident light; the first reflective axis angle is non-zero relative tothe surface normal; each of the first incident light and the secondincident light are offset from the first reflective axis; and the firstreflective axis angle differs from the second reflective axis.
 16. Theapparatus of claim 15, wherein: the first reflective axis angle differsfrom the second reflective axis angle by 0.25 degrees or less.
 17. Theapparatus of claim 15, wherein: the first internal angle of incidencediffers from the second internal angle of incidence by 20 times Δθ_(B).18. The apparatus of claim 15, wherein: the grating medium is at least70 μm thick.
 19. The apparatus of claim 15, wherein: each of the firstincident light and the second incident light are offset from the firstreflective axis by at least 1.0 degree.
 20. The apparatus of claim 15,wherein: the first incident light, the first reflected light, the secondincident light, and the second reflected light all have the samewavelength.